This curve is isomorphic to the quotient of the modular curve $X_0(91)$ by its Fricke involution $w_{91}$; this quotient is also denoted $X_0^+(91)$.
Minimal equation
Minimal equation
Simplified equation
$y^2 + (x^3 + x^2 + x + 1)y = -x^4 - 3x^3 - x^2$ | (homogenize, simplify) |
$y^2 + (x^3 + x^2z + xz^2 + z^3)y = -x^4z^2 - 3x^3z^3 - x^2z^4$ | (dehomogenize, simplify) |
$y^2 = x^6 + 2x^5 - x^4 - 8x^3 - x^2 + 2x + 1$ | (homogenize, minimize) |
Invariants
Conductor: | \( N \) | \(=\) | \(8281\) | \(=\) | \( 7^{2} \cdot 13^{2} \) | magma: Conductor(LSeries(C)); Factorization($1);
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Discriminant: | \( \Delta \) | \(=\) | \(8281\) | \(=\) | \( 7^{2} \cdot 13^{2} \) | magma: Discriminant(C); Factorization(Integers()!$1);
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Igusa-Clebsch invariants
Igusa invariants
G2 invariants
\( I_2 \) | \(=\) | \(72\) | \(=\) | \( 2^{3} \cdot 3^{2} \) |
\( I_4 \) | \(=\) | \(1236\) | \(=\) | \( 2^{2} \cdot 3 \cdot 103 \) |
\( I_6 \) | \(=\) | \(15984\) | \(=\) | \( 2^{4} \cdot 3^{3} \cdot 37 \) |
\( I_{10} \) | \(=\) | \(33124\) | \(=\) | \( 2^{2} \cdot 7^{2} \cdot 13^{2} \) |
\( J_2 \) | \(=\) | \(36\) | \(=\) | \( 2^{2} \cdot 3^{2} \) |
\( J_4 \) | \(=\) | \(-152\) | \(=\) | \( - 2^{3} \cdot 19 \) |
\( J_6 \) | \(=\) | \(392\) | \(=\) | \( 2^{3} \cdot 7^{2} \) |
\( J_8 \) | \(=\) | \(-2248\) | \(=\) | \( - 2^{3} \cdot 281 \) |
\( J_{10} \) | \(=\) | \(8281\) | \(=\) | \( 7^{2} \cdot 13^{2} \) |
\( g_1 \) | \(=\) | \(60466176/8281\) | ||
\( g_2 \) | \(=\) | \(-7091712/8281\) | ||
\( g_3 \) | \(=\) | \(10368/169\) |
Automorphism group
\(\mathrm{Aut}(X)\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(C); IdentifyGroup($1);
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\(\mathrm{Aut}(X_{\overline{\Q}})\) | \(\simeq\) | $C_2^2$ | magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
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Rational points
All points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) |
\((1 : -4 : 2)\) | \((2 : -4 : 1)\) | \((1 : -11 : 2)\) | \((2 : -11 : 1)\) |
All points | |||||
---|---|---|---|---|---|
\((1 : 0 : 0)\) | \((1 : -1 : 0)\) | \((0 : 0 : 1)\) | \((0 : -1 : 1)\) | \((-1 : -1 : 1)\) | \((-1 : 1 : 1)\) |
\((1 : -4 : 2)\) | \((2 : -4 : 1)\) | \((1 : -11 : 2)\) | \((2 : -11 : 1)\) |
All points | |||||
---|---|---|---|---|---|
\((1 : -1 : 0)\) | \((1 : 1 : 0)\) | \((0 : -1 : 1)\) | \((0 : 1 : 1)\) | \((-1 : -2 : 1)\) | \((-1 : 2 : 1)\) |
\((1 : -7 : 2)\) | \((1 : 7 : 2)\) | \((2 : -7 : 1)\) | \((2 : 7 : 1)\) |
Number of rational Weierstrass points: \(0\)
This curve is locally solvable everywhere.
Mordell-Weil group of the Jacobian
Group structure: \(\Z \oplus \Z \oplus \Z/{3}\Z\)
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -1 : 1) - (1 : -1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.600818\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.284784\) | \(\infty\) |
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0\) | \(3\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -1 : 1) - (1 : -1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0.600818\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(0\) | \(0.284784\) | \(\infty\) |
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(-z^3\) | \(0\) | \(3\) |
Generator | $D_0$ | Height | Order | |||||
---|---|---|---|---|---|---|---|---|
\((-1 : -2 : 1) - (1 : -1 : 0)\) | \(z (x + z)\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z + xz^2 - z^3\) | \(0.600818\) | \(\infty\) |
\(D_0 - 2 \cdot(1 : -1 : 0)\) | \(z^2\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z + xz^2 + z^3\) | \(0.284784\) | \(\infty\) |
\((0 : -1 : 1) - (1 : -1 : 0)\) | \(z x\) | \(=\) | \(0,\) | \(y\) | \(=\) | \(x^3 + x^2z + xz^2 - z^3\) | \(0\) | \(3\) |
BSD invariants
Hasse-Weil conjecture: | verified |
Analytic rank: | \(2\) |
Mordell-Weil rank: | \(2\) |
2-Selmer rank: | \(2\) |
Regulator: | \( 0.150828 \) |
Real period: | \( 23.53747 \) |
Tamagawa product: | \( 1 \) |
Torsion order: | \( 3 \) |
Leading coefficient: | \( 0.394457 \) |
Analytic order of Ш: | \( 1 \) (rounded) |
Order of Ш: | square |
Local invariants
Prime | ord(\(N\)) | ord(\(\Delta\)) | Tamagawa | L-factor | Cluster picture |
---|---|---|---|---|---|
\(7\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )( 1 + T )\) | |
\(13\) | \(2\) | \(2\) | \(1\) | \(( 1 - T )( 1 + T )\) |
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? |
---|---|---|
\(2\) | 2.30.2 | no |
\(3\) | 3.2160.21 | yes |
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
Splits over \(\Q\)
Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
Elliptic curve isogeny class 91.b
Elliptic curve isogeny class 91.a
Endomorphisms of the Jacobian
Of \(\GL_2\)-type over \(\Q\)
Endomorphism ring over \(\Q\):
\(\End (J_{})\) | \(\simeq\) | an order of index \(2\) in \(\Z \times \Z\) |
\(\End (J_{}) \otimes \Q \) | \(\simeq\) | \(\Q\) \(\times\) \(\Q\) |
\(\End (J_{}) \otimes \R\) | \(\simeq\) | \(\R \times \R\) |
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--Besser--Bianchi--Müller using quadratic Chabauty and the Mordell--Weil sieve.