Genus 2 curve 8281.a.8281.1

• Label: 8281.a.8281.1
• Conductor: $8281$
• Discriminant: $8281$
• Mordell-Weil group: \(\Z \oplus \Z \oplus \Z/{3}\Z\)
• Sato-Tate group: $\mathrm{SU}(2)\times\mathrm{SU}(2)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R \times \R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\Q \times \Q\)
• \(\End(J) \otimes \Q\): \(\Q \times \Q\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

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

$y^2 + (x^3 + x^2 + x + 1)y = -x^4 - 3x^3 - x^2$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(8281\)	\(=\)	\( 7^{2} \cdot 13^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(8281\)	\(=\)	\( 7^{2} \cdot 13^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(72\)	\(=\)	\( 2^{3} \cdot 3^{2} \)
\( I_4 \)	\(=\)	\(1236\)	\(=\)	\( 2^{2} \cdot 3 \cdot 103 \)
\( I_6 \)	\(=\)	\(15984\)	\(=\)	\( 2^{4} \cdot 3^{3} \cdot 37 \)
\( I_{10} \)	\(=\)	\(33124\)	\(=\)	\( 2^{2} \cdot 7^{2} \cdot 13^{2} \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_2^2$
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$C_2^2$

Rational points

All points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((0 : 0 : 1)\)	\((0 : -1 : 1)\)	\((-1 : -1 : 1)\)	\((-1 : 1 : 1)\)
\((1 : -4 : 2)\)	\((2 : -4 : 1)\)	\((1 : -11 : 2)\)	\((2 : -11 : 1)\)

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
\((-1 : -1 : 1) - (1 : -1 : 0)\)	\(z (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0.600818\)	\(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\)	\(z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0.284784\)	\(\infty\)
\((0 : -1 : 1) - (1 : -1 : 0)\)	\(z x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0\)	\(3\)

2-torsion field: 4.2.1456.1

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	Root number	L-factor	Cluster picture	Tame reduction?
\(7\)	\(2\)	\(2\)	\(1\)	\(-1\)	\(( 1 - T )( 1 + T )\)		yes
\(13\)	\(2\)	\(2\)	\(1\)	\(-1\)	\(( 1 - T )( 1 + T )\)		yes

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.