This is the submitted version of the article: Bars Cortina, Francesc; González Rovira, Josep; Xarles, Xavier. «Hyperellip- tic parametrizations of Q -curves». Ramanujan Journal, (July 2020). DOI 10.1007/s11139-020-00281-y This version is available at https://ddd.uab.cat/record/240663 under the terms of the license Hyperelliptic parametrizations of Q-curves Francesc Bars∗, Josep González †and Xavier Xarles Abstract For a square-free integer N , we present a procedure to compute Q-curves parametrized by rational points of the modular curve X0∗ (N ) when this is hyperelliptic. 1 Introduction Let X be a curve defined over Q. The curve X is said to be a Q-curve if it is isogenous to all Galois conjugates. In [Elk04], Elkies proved that every Q-curve without complex multilication (CM) is isogenous over Q to a Q-curve attached to a rational point of the modular curve X0∗ (N ) = X0 (N )/B(N ) for some square-free integer N , where B(N ) denotes the group of the Atkin-Lehner involutions of X0 (N ). Every rational non-cuspidal point in X0∗ (N ) lifts to X0 (N ) providing Q-curves, with or without CM, defined over abelian extensions of Q of type (2, · · · , 2). In [GL98], it is given a procedure to parametrize the j-invariants of these Q-curves when the ∗ genus gN of X0∗ (N ) is at most 1. In these cases, the set X0∗ (N )(Q) is infinite. Basically, in this paper it is given a method to determine the symmetric functions of the set {j(d z) : 1 ≤ d|N }, where j(z) denotes the usual generator of Q(X0 (1)), from a suitable generators of Q(X0∗ (N )). Here, we present a similar procedure for the case that X0∗ (N ) is hiperelliptic, that amounts to ∗ ∗ saying gN = 2. In fact, there are exactly 36 values of N for which gN = 2. By Faltings, for such a value of N , we are dealing with a finite number of cases. Firstly, we consider the rational points provided by Magma. Although the rank of Jac(X0∗ (N )) is equal to 2 and the classical Chabauty method does not work, we can determine the full set X0∗ (N )(Q) for 19 values of N by using a Chabauty procedure on a finite set of unramified 2-coverings of the curve. The article is organized as follows. In Proposition 1 of §2, we present the main tool to parametrize Q-curves from rational non-cuspidal points of X0∗ (N ). In §3, we give a list of equations of X0∗ (N ) when gN ∗ = 2 together their rational points provided by Magma and, in Proposition 2, we determine all rational points for 19 values of N . In §4, we show how the j-invariants of the Q-curves curves over these rational points are computed for the case N = 67. Next, for all values of N we determine which of the parametrized Q-curves have CM and, for all of them, we give the discriminant D of the order of its endomorphism ring. Moreover, if the j-invariant of the Q-curve lies in a quadratic field, it is given explicitly; otherwise, we provide the number field Q(j). We recall that there is a finite number of discriminants D of orders of imaginary quadratic fields K such that Gal(HD /Q) is of the type (2, · · · , 2), where HD denotes the ring class field of the quadratic order of discriminant D. In fact, this condition is equivalent to say that the j-invariant of an elliptic curve with CM by the order of discriminant D generates a totally real number field. Moreover, |Gal(HD /K)| divides 16 (for more detail, see [Bue89] and ∗ First and third author are supported by MTM2016-75980-P and MDM-2014-0445. † Second author is partially supported by DGI grant MTM2015-66180-R. 1 CM-computations in the web-page https://mat-web.upc.edu/people/joan.carles.lario/). The ∗ results obtained when gN = 2 show that almost these Q-curves have CM, which reinforces the conjecture that for a large enough N , the curve X0∗ (N ) does not have rational points parametrizing Q-curves without CM. 2 Preliminary results Let X be a genus two curve defined over a subfield K of the complex field C that is the normalization of the curve given by the affine equation y 2 = x 6 + a5 x 5 + a4 x 4 + a3 x 3 + a2 x 2 + a1 x + a0 , with ai ∈ K for all i ≤ 5. Let us denote by w the hyperelliptic involution. There are two points P0 and P1 = w(P0 ) over the singularity at infinity, defined over K that are not Weierstrass  y 2 y points and both satisfy (P i ) = 1. Denote by P 0 such a point satisfying (P0 ) = 1 y x3 x3 and, thus, (P1 ) = −1. For an integer n ≥ 1, we consider the K-vector space x3 Li = {f ∈ K(X) : div(f ) ≥ −n P0 } . It is clear that dim Ln = 1 for n ≤ 2 and, by the Riemann-Roch Theorem, we know that dim Ln = n−1 when n > 2. We denote by div− f the polar part of div f , i.e. div f +. div− f ≥ 0. Proposition 1. Keeping the above notation, the functions f3 , f4 , f5 ∈ K(X) defined by 8a3 − 4a4 a5 + a35 4a4 − a25 a5 1 1 f3 = + x + x 2 + x3 + y , 32 16 4 2 2 64a2 − 16a24 − 32a3 a5 + 24a4 a25 − 5a45 f4 = + x f3 , 2 256 2 3 5 128a1 − 64a3 a4 − 64a2 a5 + 48a4 a5 + 48a3 a5 − 40a4 a5 + 7a5 f5 = + x f4 , 512 vanish at w(P0 ) and satisfy that div− fi = i P0 for all i ∈ {3, 4, 5}. In particular, L3 = h1, f3 i, L4 = h1, f3 , f4 i and L5 = h1, f3 , f4 , f5 i. Proof. Let g3 ∈ L3 be a non constant function. By adding a constant, if necessary, we can assume g3 (w(P0 )) = 0. We have that the function h = g3 − (g3 |w) satisfies h|w = −h and div− h = 3(P0 ) + 3(w(P0 )). Hence, h = A y for some non zero A ∈ K. Putting f3 = g3 /A, we get f3 − (f3 |w) = y. Moreover, the functions f3 + (f3 |w) and f3 · (f3 |w) are invariant under the action of w and satisfy div− (f3 + (f3 |w)) = 3(P0 ) + 3(w(P0 )) and div− (f3 · (f3 |w)) ≤ 2(P0 ) + 2(w(P0 )) . Therefore, f3 + (f3 |w) is a polynomial in x of degree 3 and f3 · (f3 |w) must be a polynomial in x of degree at most 2. Since x6 + a5 x5 + a4 x4 + a3 x3 + a2 x2 + a1 x + a0 − (f3 + (f3 |w))2 = −4((f3 · (f3 |w)) , the polynomial P (X)2 = (f3 + (f3 |w))2 is determined and, thus, P (x) is determined up to sign. Hence, f3 must be 1/2(y ± P (x)). Since (y/x3 )(P0 ) = 1, we take the sign such that f3 = 1/2(y + x3 + · · · ). 2 The function xf3 lies in L4 and div− x f3 = 4P0 . There exists k ∈ K such that the function f4 = x f3 + k vanishes at w(P0 ). By construction, f4 − (f4 |w) = x · y and f4 + (f4 |w) is a polynomial in x of degree 4. We can determine k by using that the function x2 y 2 −(f4 +(f4 |w))2 is a polynomial in x of degree at most 3, because it is equal to the function −4((f4 · (f4 |w)). Similarly, x f4 ∈ L5 \L4 and there exists k ∈ K such that the function f5 = x f4 + k vanishes at w(P0 ). Now, f5 − (f5 |w) = x2 · y and f5 + (f5 |w) is a polynomial in x of degree 5. We determine k by using that x4 y 2 − (f5 + (f5 |w))2 is a polynomial in x of degree at most 4.  Corollary 1. For an integer n ≥ 3 and function f ∈ Ln , the function f − f (∞0 ) is a K-linear combination of the functions {f5 f3k , f4 f3k , f3k+1 : 0 ≤ k ≤ bn/3c}. Proof. For an integer n ≥ 3, let i ∈ {0, 1, 2} be such that n ≡ i (mod 3). The statement (n−i)/3−1 follows from the fact that the function h = fi+3 f3 lies in Ln with div− h = nP0 and 0 h(∞ ) = 0.  Remark 1. Note that if q is an analytic uniformizing parameter at P such that x = 1/q + · · · and y = 1/q 3 + · · · , then fi = 1/q i + · · · for i ∈ {3, 4, 5}. 3 Application to genus two curves X0∗(N ) In [HH96], it is proved that when N is square-free, X0∗ (N ) is hyperelliptic if, and only if, it has genus two. There are 35 square-free integers N such that X0∗ (N ) has genus two (cf. [Has97, B(N ) Remark 1]). In all these cases, there is an only basis h1 and P h2 of Sn 2 (Γ0 (N )) suchPthat their q-expansions lie in Z[[q]] and are of the form h1 (q) = q + n≥3 bn q and h2 (q) = q 2 + n≥3 cn q n . The functions on X0∗ (N ) defined as follows h1 dx x= = 1/q + · · · , y = −q /h2 = 1/q 3 + · · · h2 dq satisfy an equation of the form y 2 = x6 + a5 x5 + a4 x4 + a3 x3 + a2 x2 + a1 x + a0 with ai ∈ Z for all i. Denoting by ∞ the infinity cusp and by ∞0 = w(∞), we have div− x = ∞ + ∞0 , div− y = 3∞ + 3∞0 and (y/x3 )(∞) = 1. Consider the symmetric functions obtained from the functions j(d z) for 1 ≤ d|N : X Y J1 (z) = j(d z) , · · · , Jm (z) = j(d z) , 1≤d|N 1≤d|N where m = 2ω(N ) and ω(N ) denotes the number of primes dividing N . We determine every function Ji as a Q-linear combination of functions of the form f5 f3k , f4 f3k , f3k for k ≥ 0. Given a non-cuspidal Q ∈ X0∗ (N ), the j-invariants of the Q-curves attached to this point are the roots of the polynomial in z: Xn zm + (−1)i Ji (Q)z n−i . i=1 We know that for a non trivial automorphism u of X0∗ (N ), one has u(∞) 6= ∞ (cf. [BH03, Lemma 3.1]). Hence, for all these curves |X0∗ (N )(Q)\{∞}| ≥ 1 and for the biellip- tic curves, i.e. for N ∈ {106, 122, 129, 158, 166, 215, 390} (cf. [BG19, Theorem 1]), we have that | Aut(X0∗ (N ))| = 4 and, thus, |X0∗ (N )(Q)\{∞}| ≥ 3. Next, in Table 1, we present the equations with the functions x and y obtained following the procedure mentioned above, together with the rational points with x-coordinate with height less than 104 . 3 3.1 Equations and rational points N equation X0∗ (N )(Q)\{∞, ∞0 } 67 y 2 = x6 − 4x5 + 6x4 − 6x3 + 9x2 − 14x + 9 (−1, ±7), (0, ±3), (1, ±1), (2, ±1) 73 y 2 = x6 − 4x5 + 6x4 + 2x3 − 15x2 + 10x + 1 (0, ±1), (1, ±1), (2, ±3), ( 23 , ± 58 ) 85 y 2 = (x2 − 2x + 5)(x4 − 2x3 + 3x2 − 6x + 5) (0, ±5), (1, ±2), (2, ±5), ( 23 , ± 17 4 425 8 ), (− 3 , ± 27 ) 3 9 93 y 2 = (x3 − 2x2 − x + 3)(x3 + 2x2 − 5x + 3) (−1, ±3), (0, ±3), (1, ±1), (2, ±3), ( 2 , ± 8 ), ( 14 , ± 143 64 ) 103 y2 6 4 3 2 = x − 10x + 22x − 19x + 6x + 1 (0, ±1), (1, ±1), (3, ±19) 106 y2 = x6 − 4x5 + 4x4 + 2x3 + 4x2 − 4x + 1 (−1, ±4), (0, ±1), (1, ±2), (2, ±5), ( 21 , ± 58 ) 107 y2 6 5 4 3 = x − 4x + 10x − 18x + 17x − 10x + 1 2 (0, ±1), (2, ±1) 115 y2 = (x3 − 2x2 + 3x − 1)(x3 + 2x2 − 9x + 7) (1, ±1), (2, ±5), ( 12 , ± 85 ), ( 43 , ± 27 35 ) 122 y2 6 4 3 = x + 4x − 6x + 4x + 12 3 (−1, ±4), (0, ±1), (1, ±2), ( 2 , ± 8 ), ( 23 , ± 37 37 27 ) 129 y2 6 5 4 3 = x − 4x − 4x + 12x + 4x − 12x + 42 (−1, ±3), (0, ±2), (1, ±1), ( 21 , ± 38 ), (− 75 , ± 383 7 383 125 ), ( 12 , ± 1728 ) 133 y2 = x6 + 4x5 − 18x4 + 26x3 − 15x2 + 2x + 1 (0, ±1), (1, ±1), ( 35 , ± 125 83 ) 134 y2 6 5 4 3 2 = x − 4x + 2x − 2x + x + 2x + 1 (−1, ±3), (0, ±1), (1, ±1), (− 21 , 78 ) 146 y2 = x6 − 4x5 + 2x4 + 6x3 + x2 + 2x + 1 (−1, ±1), (0, ±1), (1, ±3), (2, ±5) 154 y2 = (x − 2)(x2 + x + 2)(x3 − 3x2 − x − 1) (0, ±2), (1, ±4), (2, 0), (− 23 , ± 77 8 ), (− 13 , ± 56 27 ), (4, ±22) 158 y 2 = x6 − 4x4 + 2x3 − 4x2 + 1 (0, ±1), (2, ±1), ( 12 , ± 81 ) 161 y 2 = (x3 − 2x2 + 3x − 1)(x3 + 2x2 + 3x − 5) (−1, ±7), (1, ±1), (− 21 , ± 35 1 209 8 ), (− 4 , ± 64 ) 165 y 2 = (x − 1)(x + 3)(x2 − x − 1)(x2 − x + 3) (−1, ±4), (0, ±3), (1, 0), (2, ±5), (− 12 , ± 15 8 ), (−3, 0), ( 23 , ± 5527 ), ( 5 2 , ± 99 8 ) 167 y2 = x6 − 4x5 + 2x4 − 2x3 − 3x2 + 2x − 3 (−1, ±1) 170 y2 = (x2 − 5x + 5)(x4 − 11x3 + 48x2 − 87x + 53) (1, ±2), (2, ±1), ( 32 , ± 85 ), (4, ±5), ( 11 38 3 , 27 ) 3 17 177 y2 6 4 3 2 = x + 2x − 6x + 5x − 6x + 1 (0, ±1), ( 2 , ± 8 ) 186 y2 3 2 3 2 = (x − 2x + x + 1)(x + 2x + 5x + 1) (−1, ±3), (0, ±1), (1, ±3), (2, ±9), (− 21 , ± 38 ), (− 43 , ± 143 27 ) 191 y2 6 4 3 2 = x + 2x + 2x + 5x − 6x + 1 (0, ±1), (2, ±11) 205 y2 = x6 + 2x4 + 10x3 + 5x2 − 6x + 1 (0, ±1), (−2, ±7) 206 y2 = x6 + 2x4 + 2x3 + 5x2 + 6x + 1 (−1, ±1), (0, ±1), ( 12 , ± 19 8 ) 209 y2 = x6 − 4x5 + 8x4 − 8x3 + 8x2 + 4x + 4 (0, ±2), (− 21 , ± 19 8 ) 213 y2 = x6 + 2x4 + 2x3 − 7x2 + 6x − 3 (1, ±1) 215 y2 = x6 + 4x5 − 12x4 + 20x3 − 20x2 + 12x − 4 (1, ±1), (2, ±10) 221 y2 6 5 4 3 2 = x + 4x + 2x + 6x + x − 2x + 1 (0, ±1), ( 12 , ± 98 ) 230 y2 3 2 3 = (x − 2x + 5x + 1)(x + 2x + x + 1)2 (0, ±1), (1, ±5), (−2, ±5), (3, ±35) 266 y2 = x6 + 4x5 + 10x4 + 14x3 + 17x2 + 10x + 1 (−1, ±1), (0, ±1), (− 25 , ± 83 8 ) 285 y2 2 3 2 = x(x + x + 4)(x − x − x − 3) (−1, ±4), (0, 0), (3, ±24), (− 23 , ± 57 8 ) 286 y2 3 2 3 = (x − x + 3x + 1)(x + x − 4) 2 (−1, ±4), ( 25 , ± 143 8 ) 287 y2 = x6 − 4x5 + 2x4 + 6x3 − 15x2 + 14x − 7 (−2, ±9) 299 y2 = x6 − 4x5 + 6x4 + 6x3 − 7x2 − 10x − 3 (− 12 , ± 18 ) 357 y2 6 4 3 2 = x + 8x − 8x + 20x − 12x + 12 (2, ±14) 390 y2 2 4 3 2 = (x − x + 1)(x + 5x − 8x + 5x + 1) (0, ±1), (1, ±2) Table 1 3.2 Determination of the rational points In order to determine the rational points of the curves X0∗ (N ) we will use the so-called elliptic Chabauty method, which uses a Chabauty procedure on a finite set of unramified 2-coverings of the curve. 4 Proposition 2. For the values N = 85, 93, 106, 115, 122, 129, 154, 158, 161, 165, 170, 186, 209, 215, 230, 285, 286, 357 and 390, the set of rational points of X0∗ (N )(Q) is the set given in Table 1 together with the two points at infinity. Proof. The proposition is proved by using some computations in MAGMA [BP97]. A file with all the computations can be downloaded from the github account of the third author. We explain the main ideas in the computation, that can be done for any hyperelliptic curve X of genus g (g = 2 in our cases) given by an equation of the form y 2 = f (x), whit deg f (x) = 2g +2. The computation is done in two steps: first one computes the finite set of twists Cξ of the unramified coverings of the curve X with Galois group ∼ = (Z/2Z)2g which have points locally for any prime p; this is completely analogous to the 2 descent for elliptic curves, as described in [BS09]. Each twists Cξ is associated to an element ξ ∈ (Q[x]/f (x))∗ (where the twist corresponding the points at infinity corresponds to ξ = 1). If some of the curves does not have rational points (apparently), one needs to show this by using either a Mordell-Weil sieve or a higher descent. For our curves this never happens, so we will not analyze this case further. Our aim now is the determination of the rational points in Cξ (Q). Now, the jacobian of any of this curves has quotients isomorphic to the Weil restriction of elliptic curves Eξ defined over some number fields K, and the rational points in Cξ (Q) give points in Eξ (K) whose image with respect to a given map ϕξ : Eξ → P1 is in P1 (Q); this is the necessary data for the elliptic Chabauty function, which computes the set of points in Eξ (K) verifying this condition if rankZ (Eξ (K)) < deg(K/Q). In practice, the fields K we need are the minimal field of definition of some fixed factorization f (x) = g(x)h(x) where g(x) has degree 4. For example, if g = 2 and the polynomial f (x) is irreducible, we consider the field L0 := Q[x]/f (x). Suppose furthermore that f (x) = (x − α)f1 (x) in L[x] again with f1 (x) irreducible; then the minimal field of definition of a factorization f (x) = g(x)h(x) where g(x) has degree 4 is a field of degree 15 over Q. In this case, the elliptic curves correspond to the jacobians of the curves Hξ : y 2 = ξg(x), and the map ϕξ : Hξ → P1 is given by the x-coordinate. In this case the necessary Chabauty condition is rankZ (Eξ (K)) < 15, which is quite likely to be fulfilled; but right now the computation of the rank and a finite index subgroup of E(K) for K/Q of such degree is unfeasible. This situation is what happens in all the values of N which we were not able to determine the set of rational points (including all prime values of N ). The other extreme case is when there is a factorization f (x) = g(x)h(x) already defined over Q; in this case we need to compute the rational points of the curves y 2 = dξ g(x) for some values dξ ∈ Q∗ , which are only finite if the rank is zero (which is very unlikely to happen for all the necessary twists dξ ). The best cases are when one can find such a factorization over a field of degree ≤ 4 for any twists verifying the corresponding Chabauty condition. In some cases we used distinct fields for different twists, as we explain in the following example. Example. We explain in detail the case X0∗ (85), where f (x) = (x2 −2x+5)(x4 −2x3 +3x2 −6x+5). We have 3 17 4 425 X0∗ (85)(Q) = {(0, ±5), (1, ±2), (2, ±5), ( , ± ), (− , ± ), ±∞}. 2 8 3 27 We have 5 twists, corresponding to the x-coordinates of the rational points, except that the points with x = 1 already appear in the trivial twists corresponding to the points at infinity. If we consider the given factorization over Q, the corresponding elliptic curves for all the twists except the trivial one have rank 1. On the other hand the curve H given by the equation y 2 = x4 − 2x3 + 3x2 − 6x + 5 has rank 0 and 4 points, corresponding to the points at infinity and the points with x = 1. If we adjoint a root of x2 − 2x + 5 we get a quadratic extension K2 /Q. Over this extension we get also a factorization of x4 − 2x3 + 3x2 − 6x + 5 = h1 (x)h2 (x) with deg(hi (x)) = 2 for 5 i = 1, 2. So we can take f (x) = g(x)h1 (x) for some g(x) of degree 4. The twists corresponding to the points with x-coordinate 32 and − 34 have rank 1, and Chabauty method succeeds. But the ones corresponding to the points with x-coordinate 0 and 2 have rank 2. If we adjoint a root of x4 − 2x3 + 3x2 − 6x + 5 we get a field K4 where f (x) has 4 roots and a degree 2 factor. By considering the corresponding degree 4 polynomial as a product of two (adequate) degree one factors and the degree two factor we finally get that the remaining twist have jacobian of rank 1 and we find a non-torsion point in each case, and Chabauty computations succeeds. There is one case were the approach described above did not succeed. Example. In the case X0∗ (390) we had to do a slightly modified approach; in fact, we tried fields of degree 1 and 2 and the Chabauty condition was not fulfilled, and we had to go to a degree 8 extension, where the rank computation did not succeed. Instead, we showed that all the rational points in X0∗ (390) : y 2 = (x2 − x + 1)(x4 + 5x3 − 8x2 + 5x + 1) came from the trivial twists over Q. This means that their x-coordinates verify that y12 = x2 − x + 1 and y22 = x4 + 5x3 − 8x2 + 5x + 1 for some y1 , y2 ∈ Q. The curve X 0 determined by these equations is an hyperelliptic curve of genus 3, whose hyperelliptic equation can be computed by parametrizing the first equation. We get the equation X 0 : z 2 = t8 + 10t7 − 41t6 + 42t5 + 33t4 − 76t3 + 44t2 − 16t + 4. Now we apply the above method to this new curve X 0 . The curve has (apparently) 12 rational points (two above each rational point of X0∗ (390), as it is an unramified 2-covering), with 6 possible values for the x-coordinates. √ We computed there are exactly 6 possible twists, one for each x-coordinate. Over K = Q( 5) the defining hyperelliptic polynomial factors as a product of two degree 4 polynomials. For every twists one of the two quotient elliptic curves of the corresponding covering has rank one, and the Chabauty computation succeeds.  4 An example and results First, we show, for the case N = 67, the procedure used. The equation obtained in Table 1 is y 2 = x6 − 4x5 + 6x4 − 6x3 + 9x2 − 14x + 9. Let j67 (q) := j(q 67 ) = 1/q 67 + 744 + · · · and put J1 = j + j67 and J2 = j · j67 . With the notation of before section, we have 1 −1 + x − 2x2 + x3 + y ,  f3 = f4 = x f3 + 1 , f5 = x f4 − 1 . 2 After computing, we obtain J1 = −23f322 + f321 f4 − 1279f321 − 781f320 f4 + 186f320 f5 − 99914f320 − 39399f319 f4 + 14954f319 f5 − 2698696f319 − 265633f318 f4 + 380472f318 f5 − 20514523f318 + 6929641f317 f4 + 1576893f317 f5 − 49240824f317 + 67627402f316 f4 − 16450546f316 f5 − 61401116f316 + 190686364f315 f4 − 81315034f315 f5 − 56264079f315 + 259977664f314 f4 − 148558638f314 f5 − 88533538f314 + 95806265f313 f4 − 69608123f313 f5 − 162557463f313 − 295479289f312 f4 + 158123161f312 f5 − 27169544f312 − 558873206f311 f4 + 260674425f311 f5 + 456803156f311 − 423722114f310 f4 + 202709065f310 f5 + 731171796f310 − 51627779f39 f4 + 133780373f39 f5 + 234273070f39 + 264268555f38 f4 − 64460559f38 f5 − 502745764f38 + 337312727f37 f4 − 318069668f37 f5 − 623447279f37 + 158991342f36 f4 − 299948399f36 f5 − 229889114f36 − 28504966f35 f4 − 91453878f35 f5 + 60433254f35 − 60041832f34 f4 + 24362830f34 f5 + 79628320f34 − 20570848f33 f4 + 26593344f33 f5 + 23436576f33 − 941600f32 f4 + 8600928f32 f5 + 1047456f32 + 647200f3 f4 + 1386464f3 f5 − 571936f3 + 81536f4 + 92000f5 − 65536 , 6 and J2 = f5 f321 + 720f4 f321 + 179980f322 + 17300122f5 f320 + 410510311f4 f320 + 5149868567f321 + 42380978353f5 f319 + 176848900626f4 f319 + 839384847849f320 + 3347632163474f5 f318 + 5490095012794f4 f318 + 20232433296285f319 + 58903607428273f5 f317 + 52642797600751f4 f317 + 194740838424278f318 + 465038243745693f5 f316 + 252244831282013f4 f316 + 1049935814900775f317 + 2137932776610224f5 f315 + 718160756825707f4 f315 + 3682889678423580f316 + 6483265127099687f5 f314 + 1296256091753552f4 f314 + 9112585010431660f315 + 13905322982585428f5 f313 + 1455284181206971f4 f313 + 16698439687036025f314 + 22013212259613947f5 f312 + 785970239416766f4 f312 + 23380324317471236f313 + 26434965501463926f5 f311 − 389286347179143f4 f311 + 25533239990035759f312 + 24511794195150313f5 f310 − 1257772458271356f4 f310 + 22044986048091002f311 + 17745745315071401f5 f39 1369085873848977f4 f39 + 15174609069161127f310 + 10090414014987393f5 f38 − 954179629348608f4 f38 + 8365619455927489f39 + 4511870660986624f5 f37 − 476646738132432f4 f37 + 3699701950122944f38 + 1580542220676176f5 f36 − 178202666014272f4 f36 + 1312789289946672f37 + 429477284579616f5 f35 − 51157416030048f4 f35 + 374258702810464f36 + 88841374862944f5 f34 − 11506808706816f4 f34 + 86310527850080f35 + 13536732334080f5 f33 − 2038076477952f4 f33 + 16327740526336f34 + 1432454117376f5 f32 − 272275906560f4 f32 + 2547759661312f33 + 93842541824f5 f3 − 24005742848f4 f3 + 313280547584f32 + 2852000000f5 − 1009741824f4 + 26269884672f3 + 1073741824 . Hence, the j-invariants of the Q-curves attached to ∞0 are the solutions of the equation z 2 − J1 (∞0 ) z + J2 (∞0 ) = 0 . Since (J1 (∞0 ), J2 (∞0 )) = (−65536, 1073741824), we get j = −323 , which corresponds to an elliptic curve with CM by the quadratic order of discriminant −11. The remaining rational non-cuspidal points provide Q-curves with CM: point j point j point j point j (−1, 7) 2553 3 (0, 3) −3 · 160 (1, 1) 203 (2, 1) −9603 (−1, −7) −52803 (0, −3) 0 (1, −1) −153 (2, −1) 2 · 303 Next, we show the results obtained. 4.1 N is a prime: Gal(Q(j)/Q) ,→ Z/2Z. N point CM D j-invariant 67 ∞0 yes −11 −323 (−1, 7) yes −7 2553 (−1, −7) yes −67 −52803 (0, 3) yes −27 −3 · 1603 (0, −3) yes −3 0 (1, 1) yes −8 203 (1, −1) yes −7 −153 (2, 1) yes −43 −9603 (2, −1) yes −12 2 · 303 73 ∞0 yes −12 2 · 303 (0, 1) yes −27 −3 · 1603 (0, −1 yes) −4 123 (1, 1) yes −19 −963 (1, −1) yes −8 203 (2, 3) yes −67 −52803 (2, −3) yes −16 663 (3/2, 5/8) yes −3 0  √ 3 3(−26670989 ± 15471309 −127) (3/2, −5/8) non − 20 226 7 N point CM D j-invariant 103 ∞0 yes −67 −52803 (0, 1) yes −43 −9603 (0, −1) yes −27 −3 · 1603 (1, 1) yes −19 −963 ] 3 (1, −1) yes −12 √ 2 · 303 (3, 19) non − 19(48(1623826405 ± 30228849 2885)) (3, −19) yes −3 0 107 ∞0 yes −8 203 (0, 1) yes −7 −153 (0, −1) yes −43 −9603 (2, 1) yes −67 −52803 (2, −1) yes −28 2553 167 ∞0 yes −43 −9603 (−1, 1) yes −67 −52803 (−1, −1) yes −163 −6403203 191 ∞0 yes −43 −9603 (0, 1) yes −11 −323 (0, −1) yes −7 −153 (2, 11) non − j0 (2, −11) yes −28 2553 where √ !3 √ 480(7725788647437 ± 95942438 2036079533) j0 = (724537954586714121 ± 16056976492100 2036079533) 1912 4.2 N is a product of two primes: Gal(Q(j)/Q) ,→ (Z/2Z)2 . N point CM D j or Q(j) 0 85 ∞ yes −19 √−963 (0, 5) yes −35 −(16(15 √ 3 ±√ 7 5))3 (0, −5) yes −60 (3(470 ± 213 5) (1 ± 5)/2) (1, 2) yes −16 663 3 (1, −2) yes −4 √ 123 (2, 5) yes −115 −(48(785 √ ± 351√ 5)) 3 (2, −5) yes −15 −(3(25 ± 9 5)/2) √ (−1 ± √5)/2 (3/2, 17/8) yes −51 −(48(37 ± 9 17))3√ (−4 ± √ 17) (3/2, −17/8) non − √ 17, Q( √ −95) (−4/3, 425/27) non − Q( 85, √−4295) √ (−4/3, −425/27) yes −595 Q( 5, 17) 8 N point CM D j or Q(j) 0 3 93 ∞ yes −12 √ 3 √2 · 30 (0, 3) yes −60 (3(470 ± 213 √5) (1 ± 5)/2) √ 3 (0, −3) yes −24 √± 2 3 2)) (3 ± √ (12(5 2 2) (−1, 3) yes −123 −(480(461 ± 72 41)) (−32 √ ±3 5 √41) (−1, −3) yes −75 −(48(−69 ± 31 5)) (± 5) (1, 1) yes −11 −323 (1, −1) yes −3, −12 √ 0, −3 ·√1603 3 (2, 3) yes −147 −3(480(142 ± √31 3 21)) (±√ 21) (2, −3) yes −15 −(3(−5 ± 4 5)) (−3 ± √5)/2 (3/2, 9/8) yes −48 √ ±√ 4(15(30 17 3))3 (3/2, −9/8) non − Q( −15, −109) (1/4, 143/64) yes −3 √ √ 0 (1/4, −143/64) non − Q( −23, −143) 106 ∞0 yes −7 √ 3 −15 √ 3 (−1, 4) yes −36 4(21 ± 20 3) (7 ± √4 3)3 (−1, −4) yes −148 (60(2837 ± 468 √37)) (0, 1) yes −40 (6(65 ± 27 5))3 3 (0, −1) yes −4, −16 √ 3 12 ,√ 663 (1, 2) yes −24 (12(9 ± 7 2)) (−1√ ± 2) (1, −2) yes −52 (30(31 ± 9 √13))3 (2, 5) yes −100 (6(2927 ± 1323 5))3 3 (2, −5) yes −4 √ √ 12 (1/2, 5/8) non − Q( 33, −591) (1/2, −5/8) yes −7, −28 −153 , 2553 0 √ 115 ∞ yes −115 (48(−785 ± 351 5))3 (1, 1) yes −19 −963 (1, −1) yes −11 √−323 3 (2, 5) yes −235 √ 3 ± 3969√ 5)) (528(−8875 (2, −5) yes −15 −(3/2(25 ± 9 5))√(−1√ ± 5)/2 (1/2, 5/8) non − Q( 65, −3495) √ 3 (1/2, −5/8) yes −40 (6(65 √ 3 ± 27 √ 5)) (4/3, 35/27) yes −60 (3(470 ± 213 5) ± √ √ 5)/2) (1 (4/3, −35/27) non − Q( 10, −9278) √ √ 122 ∞0 yes −36 −(4(102 ± 61 3))3 (−2√ ± 3) (−1, 4) yes −52 (30(31 ± 9 √13))3 (−1, −4) yes −100 (6(2927 ± 1323 5))3 (0, 1) yes −3, −12 0, 2 · 303 3 (0, −1) yes −4, −16 12√ , 663 (1, 2) yes −88 (60(155 ± 108√2))3 (1, −2) yes −20 (2(25 ± 13√ 5))3 3 (3/2, 37/8) yes −232 (30(140989 ± √26163√ 29)) (3/2, −37/8) non − Q( −15, 1585) (2/3, 37/27) yes −4 √ √ 123 (2/3, −37/27) non − Q( 1258, −1598) 9 N point CM D j or Q(j) √ 3 √ 129 ∞0 yes −75 −(48(69 √ ± 313 5)) (±√ 5) (−1, 3) yes −123 −(480(−461 ± 72 41)) (32 ± 5√ 41) 3 (−1, −3) yes −48 √ ±317 √3)) 4(15(30 (0, 2) yes −147 −3(480(362 ± 79 21)) (± 21) (0, −2) yes −8 203 (1, 1) yes −3, −27 0, −3 · 1603 (1, −1) yes −12 √ 2√· 303 3 (1/2, 3/8) yes −51 −(48(37 ± 9 17))√ (−4√ ± 17) (1/2, −3/8) non − Q( 57, −687) (−7/5, 383/125) yes −3 √ √ 0 (−7/5, −383/125) non − Q(√ 1149, √ −1059) (7/12, 383/1728) non − Q( −7, √ −444783) √ (7/12, −383/1728) non − Q( 85, −347) √ √ 133 ∞0 non − Q( 2, 69) (0, 1) yes −27 −3 · 1603 3 (0, −1) yes −19 √−963 (1, 1)) yes −91 (48(−227 ± 63 13)) (1, −1) yes −12 2 · 303 (3/5, 83/125) yes −3 √ √ 0 (3/5, −83/125) non − Q( −31, −3651) √ 134 ∞0 yes −52 (30(31 ± 9 13))3 3 (−1, 3) yes −7 √−153 (−1, −3) yes −232 (30(140989 ± 26163 √29)) (0, 1)) yes −20 (2(25 ± 13 5))3 (0, −1) yes −3, −12 0, 2 · 303 (1, 1) yes −8 203 2 3 (1, −1) yes −7, −28 √ −15 √ , 255 (−1/2, 7/8) non − √ 113, Q( −1271) √ 3 (−1/2, −7/8) yes −72 (20(389 ± 158 6)) (−5 ± 2 6) 146 ∞0 yes −3, −12 √ 3 0, 2 ·√ 303 (−1, 1) yes −36 −(4(102 ± 61 3)) (−2√ ± 3) (−1, −1) yes −148 (60(2837 ± 468 37))3 3 (0, 1)) yes −4, −16 √ 3 12 ,√ 663 (0, −1) yes −24 (12(9 ± 7 2)) (−1 ± 2) (1, 3) yes −8 √ 3 √203 (1, −3) yes −72 −(20(389 ± 158 6)) (−5 ±√ 2 6) (2, 5) yes −100 (6(2927 ± 1323 5))3 (2, −5) yes −4 123 158 ∞0 yes −7 −153 (0, 1) yes −3, −12 √ 3 0, 2 ·√ 303 (0, −1) yes −24 (12(9 ± 7 2)) (−1√ ± 2) (2, 1)) yes −232 (30(140989 ± 26163√29))3 3 (2, −1) yes −148 √ ± 468 (60(2837 √ 37)) (1/2, 1/8) non − Q( 1169, −1247) (1/2, −1/8) yes −7, −28 −153 , 2553 10 N point CM D j or Q(j) 0 3 161 ∞ yes −7 √−153 (−1, 7) yes −91 −(48(227 ±√63 √ 13)) (−1, −7) yes −483 Q( 21,√ 69) (1, 1)) yes −115 −(48(785 ± 351 5))3 3 (1, −1) yes −19 √ √ −96 (−1/2, 35/8) non Q( −7, 32009)√ (−1/2, −35/8) yes −112 (15(2168 ± 819 7))3 (−1/4, 209/64) yes −8 √ √ 2553 (−1/4, −209/64) non Q)( 209, −1140391) 0 177 ∞ yes − − 11 √ 3 √323 (0, 1) yes −24 (12(9 ± 7 2)) (−1 ± 2) (0, −1) yes −8 √ √ 203 (3/2, 17/8) yes −267 −(240(562501 ± 59625 89))3 (−500 √ ± 53 √ 89) (3/2, −17/8) non − Q( −23, 2881) 0 √ 205 ∞ yes −115 −(48(785 ± 351 5))3 3 (0, 1) yes −16 √ 663 (0, −1) yes −40 (6(65 ± 27 5)) 3 (−2, 7) yes −4 √ √12 (−2, −7) yes −1435 Q( 5, 21) 0 √ 3 √ 206 ∞ yes −24 (12(9 ± 7 2)) (−1 ± 2) (−1, 1) yes −3, −12 0, 2√· 303 (−1, −1) yes −88 (60(155 ± 108 √ 2))3 (0, 1) yes −40 6(65 ± 27√ 5)3 (0, −1) yes −20 (2(25 ± 13√ 5)) (1/2, 19/8) yes −148 √ ±√ (60(2837 468 37))3 (1/2, −19/8) non − Q( 193, −27119) 0 209 ∞ yes −8 203 (0, 2) yes −19 √−963 (0, −2) yes −88 (60(155 √ ±√108 2))3 (−1/2, 19/8) non − Q( −1007,√ 902537)√ (−1/2, −19/8) yes −627 Q( 33, 57) √ √ 213 ∞0 −51 yes −(48(37 ±√9 17))3 (−4 ± √17) (1, 1) yes −123 (480(461 ± 72 41))3 (−32 ± 5 41) (1, −1) yes −11 −323 √ √ 215 ∞0 non − Q( 2, 47645) √ (1, 1) yes −235 −(528(8875 ± 3969 5))3 (1, −1) yes −19 √ √ −963 (2, 10) non − Q( 85, 3418805)√ (2, −10) yes −115 −(48(785 ± 351 5))3 221 ∞0 yes −16 663 3 (0, 1) yes −43 √ −960 √ (0, −1) yes −51 −(48(37 ± 9√ 17))3 (−4 √ ± 17) (1/2, 9/8) non − Q( 1081, −779263) (1/2, −9/8) yes −4 123 11 N point CM D j or Q(j) √ 287 ∞0 yes −91 −(48(227 ± √ 63 √ 13))3 (−2, 9) yes −1435 √ √ Q( 5, 41) (−2, −9) non − Q( 8321, 2904137173) √ 299 ∞0 yes −91 −(48(227 ± 63 13))3 (−1/2, 1/8) yes −43 √ √ −9603 (−1/2, −1/8) non Q( 1513, −3325543) 4.3 N is a product of three primes: Gal(Q(j)/Q) ,→ (Z/2Z)3 . N point CM D j or Q(j) √ 154 ∞0 yes −40 (6(65√± 27 5))3 √ (0, 2) yes −24 (12(9 ± 7 2))3√ (−1 ± 2) (0, −2) yes −52 (30(31 ± 9 13))3 (1, 4) yes −7 −153 (1, −4) yes −7, −28 −15 3 3 √ , 255 √ (2, 0) yes −84 √ Q(√3, 7) √ (−3/2, 77/8) non −− Q( −143, √ √ −185, √ −455) (−3/2, −77/8) yes −1540 √Q( √5, 7, √11) (−1/3, 56/27) non −− Q( 7, 5 · 11, −479) √ 3 (−1/3, −56/27) yes −28, −112 2553 , (15(2168 √ √ √ ± 819 7)) (4, 22) yes −1848 Q( 2,√ 21, √ 33) (4, −22) yes −132 Q( 3, 11) 165 ∞0 yes −11 −32√ √ 3 (0, 3) yes −195 Q(√ 5, 13) √ (0, −3) yes −51 (48(37 ± 9 √17))3 (−4 ±√ 17) (1, 0) yes −24 (12(9 ± 7 √2))√ 3 (−1 ± 2) (2, 5) yes −435 Q( 5, 29) √ 3 (2, −5) yes −35 −(16(15 √ √ ± 7 √5)) (−1/2, 15/8) non −− Q( −15,√ 265, √ 1745) (−1/2, −15/8) yes −120 √ √ 5) Q( 2, √ (−3, 0) yes −1155 Q( 5, √ 21, 33) √ (2/3, 55/27) yes −11, −99 −323 , (16(3751√ ± 653 √ 33)) 3 √ (−23 ± 4 33) (2/3, −55/27) non −− Q( −11, √ √47,√ −661) (5/2, 99/8 yes −1320 √ 5, √ 6, √22) Q( (5/2, −99/8) non −− Q( −7, 33, 393) √ √ 170 ∞0 yes −36 −(4(102 ± 61 3))3 (−2 ± 3) (−1, 2) yes −4, −16 123 3 √ , 66 √ (−1, −2) yes −340 Q( 5, 17) √ (0, 1) yes −4, −100 123 , (6(2927 √ ± 1323 5)) √ 3 (0, −1) −15 −(3(25 ± 9 5)/2) 3 yes √ √(−1 ± 5)/2 (2, 5) yes −280 √ √Q( 2, 5) √ 3 √ (2, −5) 3 yes −15, −60 −(3(25 ± 9 5)/2) (−1 ± 5)/2, −(3(470 ± 213 5)) (1 ± 5)/2 √ √ (−1/2, 5/8) yes −120 Q( √ √ 2, 5)√ (−1/2, −5/8) non −− Q( √17, √−95,√ 65) (5/3, 38/27) non −− Q( 73, 19, −5) (5/3, −38/27) yes −4 123 12 N point CM D j or Q(j) 186 ∞ 0 yes −3, −12 303 0,√2 · √ (−1, −3) yes −228 Q(√ 3, 319) √ (0, 1) yes −15 −(3(25 ± 9 √ 5)/2) (−1 ±√ 5)/2 (0, −1) yes −24 (12(9 ± 7 √2))√ 3 (−1 ± 2) (1, 3) yes −168 Q( √6, √14) (1, −3) yes −120 Q(√ 2,√ 5) (2, 9) yes −708 √ √Q( 3, 59) √ 3 √ (2, −9) yes −15, −60 −(3(25 ± 9 5)/2)3 (−1 √± 5)/2, √ −(3(470 √ ± 213 5)) (1 ± 5)/2 (−1/2, 3/8) non − Q( −15, 177, 1257) √ (−1/2, −3/8) yes −12, −48 2 · 30 3 , 4(15(30 ± 17 3))3 √ √ √ (−4/3, 143/27) non − Q( 37, √−143, √ 2077) (−4/3, −143/27) yes −332 Q( 3, 31) √ 230 ∞0 yes −40 (6(65 ± 27√5))3 −20 3 (0, 1) yes (2(25 √ ± 133 5)) √ (0, −1) yes −15 −(3(25 ± 9 √ 5)/2)√ (−1 ± 5)/2 (1, 5) yes −520 Q( √5, √13) (1, −5) yes −120 √ √Q( 2, 5) √ √ 3 −15, −60 −(3(25 ± 9 5)/2) (−1 ± √ ± 213 5))3 (1 ± 5)/2 (−2, 5) yes 5)/2,√ −(3(470 √ (−2, −5) yes −1380 √Q( 3, √ 5, √23) (3, 35) non − Q( 685, √705, √ 19043) (3, −35) yes −180 Q( 3, 5) 0 √ 266 ∞ yes −52 (30(31√± 9√ 13))3 (−1, 1) yes −84 Q( 3, 7) (−1, −1) yes −3, −12 0,√2 · 30 3 √ (0, 1) yes −280 Q( 2, √5) (0, −1) yes −40 (6(65 ± 3 √ √ 27 5))√ (−5/2, 83/8) non − Q( 1041, √ −415,√ 105) (−5/2, −83/8) yes −532 Q( 17, 19) 0 √ √ 285 ∞ yes −51 −(48(37 ±√ 9 17))3 (−4 ±√ 17) (−1, 4) yes −15 −(3(25 ± 9 5)/2) 3 ± √ 5)/2 √ (−1 3 (1 ± 5)/2 (−1, −4) yes −60 −(3(470 ± 213 5)) √ 3 √ (0, 0) yes −3, −75 0, −(48(−69 √ √± 31√ 5)) (± 5) (3, 24) non − Q( 3, √95,√ 60197) (3, −24) yes −240 √ Q( √3, √ 5) (−3/2, 57/8) non Q( −79, √ √ √11985) 57, (−3/2, −57/8) yes −1995 Q( 5, 21, 57) 0 √ 286 ∞ yes −40 (6(65 ± 27√ 5))3 (−1, 4) yes −52 (30(31 ± 9 13)) 3 √ 3 (−1, −4) yes −88 √ √ ± 108 √ (60(155 2)) (5/2, 143/8) non − Q( √ 39, √ 168917, √ 232) (5/2, −143/8) non − Q( 1841, −3367, 37609) 0 √ √ 357 ∞ yes −168 √ Q( √6, 14) √ (−1, 4) non − Q( 293, 89997, √ 321) (−1, −4) yes −35 −(16(15 ± 7 5)) 13 4.4 N is a product of four primes: Gal(Q(j)/Q) ,→ (Z/2Z)4 . 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[GL98] Josep González and Joan-C. Lario. Rational and elliptic parametrizations of Q-curves. J. Number Theory, 72(1):13–31, 1998. [Has97] Yuji Hasegawa. Hyperelliptic modular curves X0∗ (N ). Acta Arith., 81(4):369–385, 1997. [HH96] Yuji Hasegawa and Ki-ichiro Hashimoto. Hyperelliptic modular curves X0∗ (N ) with square-free levels. Acta Arith., 77(2):179–193, 1996. Francesc Bars Cortina Departament Matemàtiques, Edif. C, Universitat Autònoma de Barcelona 08193 Bellaterra, Catalonia francesc@mat.uab.cat Josep González Rovira Departament de Matemàtiques, Universitat Politècnica de Catalunya EPSEVG, Avinguda Vı́ctor Balaguer 1, 08800 Vilanova i la Geltrú, Catalonia josep.gonzalez@upc.edu 14 Xavier Xarles Departament Matemàtiques, Edif. C, Universitat Autònoma de Barcelona 08193 Bellaterra, Catalonia xarles@mat.uab.cat 15