This is a model for the quotient of the modular curve $X_0(67)$ by its Fricke involution $w_{67}$; this quotient is also denoted $X_0^+(67)$. This is the first genus-$2$ modular curve $X_0(N)/\langle w_N \rangle$ for $N$ prime.
Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x + 1)y = x^5 - x$ | (homogenize, simplify) |
$y^2 + (x^3 + xz^2 + z^3)y = x^5z - xz^5$ | (dehomogenize, simplify) |
$y^2 = x^6 + 4x^5 + 2x^4 + 2x^3 + x^2 - 2x + 1$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(4489\) | \(=\) | \( 67^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
|
Discriminant: | \( \Delta \) | \(=\) | \(4489\) | \(=\) | \( 67^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
|
Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(284\) | \(=\) | \( 2^{2} \cdot 71 \) |
\( I_4 \) | \(=\) | \(1369\) | \(=\) | \( 37^{2} \) |
\( I_6 \) | \(=\) | \(127187\) | \(=\) | \( 193 \cdot 659 \) |
\( I_{10} \) | \(=\) | \(-574592\) | \(=\) | \( - 2^{7} \cdot 67^{2} \) |
\( J_2 \) | \(=\) | \(71\) | \(=\) | \( 71 \) |
\( J_4 \) | \(=\) | \(153\) | \(=\) | \( 3^{2} \cdot 17 \) |
\( J_6 \) | \(=\) | \(187\) | \(=\) | \( 11 \cdot 17 \) |
\( J_8 \) | \(=\) | \(-2533\) | \(=\) | \( - 17 \cdot 149 \) |
\( J_{10} \) | \(=\) | \(-4489\) | \(=\) | \( - 67^{2} \) |
\( g_1 \) | \(=\) | \(-1804229351/4489\) | ||
\( g_2 \) | \(=\) | \(-54760383/4489\) | ||
\( g_3 \) | \(=\) | \(-942667/4489\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
|
\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
|
Rational points
All points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) |
\((-1 : 1 : 1)\) | \((1 : -3 : 1)\) | \((1 : -3 : 2)\) | \((1 : -10 : 2)\) |
All points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((-1 : 0 : 1)\) | \((0 : -1 : 1)\) | \((1 : 0 : 1)\) |
\((-1 : 1 : 1)\) | \((1 : -3 : 1)\) | \((1 : -3 : 2)\) | \((1 : -10 : 2)\) |
All points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) |
\((1 : -3 : 1)\) | \((1 : 3 : 1)\) | \((1 : -7 : 2)\) | \((1 : 7 : 2)\) |
Number of rational Weierstrass points: \(0\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z \oplus \Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(2 \cdot(0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.133201\) | \(\infty\) |
\((0 : 0 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.096171\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(2 \cdot(0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.133201\) | \(\infty\) |
\((0 : 0 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.096171\) | \(\infty\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\(2 \cdot(0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) | \(x^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 - z^3\) | \(0.133201\) | \(\infty\) |
\((0 : 1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + xz^2 + z^3\) | \(0.096171\) | \(\infty\) |
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.011438 \) |
Real period: | \( 20.46511 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 1 \) |
Leading coefficient: | \( 0.234099 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(67\) | \(2\) | \(2\) | \(1\) | \(( 1 + T )^{2}\) |
Galois representations
For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.
For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.
Prime \(\ell\) | mod-\(\ell\) image | Is torsion prime? |
---|---|---|
\(2\) | 2.12.2 | no |
\(3\) | 3.2160.18 | no |
Sato-Tate group
\(\mathrm{ST}\) | \(\simeq\) | $\mathrm{SU}(2)\times\mathrm{SU}(2)$ |
\(\mathrm{ST}^0\) | \(\simeq\) | \(\mathrm{SU}(2)\times\mathrm{SU}(2)\) |
Decomposition of the Jacobian
Simple over \(\overline{\Q}\)
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | \(\Z [\frac{1 + \sqrt{5}}{2}]\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q(\sqrt{5}) \) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).
Additional information
The set of rational points on this genus 2 curve with rank 2 Jacobian was computed by Balakrishnan--Best--Bianchi--Lawrence--Müller--Triantafillou--Vonk using quadratic Chabauty and the Mordell--Weil sieve.