Genus 2 curve 869526.a.869526.1

• Label: 869526.a.869526.1
• Conductor: $869526$
• Discriminant: $869526$
• Mordell-Weil group: \(\Z \oplus \Z \oplus \Z/{2}\Z\)
• Sato-Tate group: $\mathrm{USp}(4)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\Q\)
• \(\End(J) \otimes \Q\): \(\Q\)
• \(\overline{\Q}\)-simple: yes
• \(\mathrm{GL}_2\)-type: no

Minimal equation

$y^2 + (x^3 + 1)y = 10x^5 + 98x^4 + 64x^3 - 233x^2 + 133x - 23$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(869526\)	\(=\)	\( 2 \cdot 3^{2} \cdot 7 \cdot 67 \cdot 103 \)
Discriminant:	\( \Delta \)	\(=\)	\(869526\)	\(=\)	\( 2 \cdot 3^{2} \cdot 7 \cdot 67 \cdot 103 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(3558964\)	\(=\)	\( 2^{2} \cdot 41 \cdot 21701 \)
\( I_4 \)	\(=\)	\(140093089\)	\(=\)	\( 1093 \cdot 128173 \)
\( I_6 \)	\(=\)	\(165658364763297\)	\(=\)	\( 3 \cdot 7^{5} \cdot 1601 \cdot 2052157 \)
\( I_{10} \)	\(=\)	\(111299328\)	\(=\)	\( 2^{8} \cdot 3^{2} \cdot 7 \cdot 67 \cdot 103 \)

Automorphism group

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

Rational points

Known points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (1 : -4 : 2),\, (1 : -5 : 2),\, (29 : -120189 : 60),\, (29 : -120200 : 60)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z \oplus \Z \oplus \Z/{2}\Z\)

Generator	$D_0$						Height	Order
\(2 \cdot(1 : -5 : 2) - (1 : -1 : 0) - (1 : 0 : 0)\)	\((2x - z)^2\)	\(=\)	\(0,\)	\(4y\)	\(=\)	\(-19xz^2 + 7z^3\)	\(1.419727\)	\(\infty\)
\((1 : -5 : 2) - (1 : 0 : 0)\)	\(z (2x - z)\)	\(=\)	\(0,\)	\(2y\)	\(=\)	\(-2x^3 - z^3\)	\(0.407525\)	\(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x^2 + 14xz - 7z^2\)	\(=\)	\(0,\)	\(2y\)	\(=\)	\(-203xz^2 + 97z^3\)	\(0\)	\(2\)

2-torsion field: 6.6.75271512927744.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(5\)
Regulator:	\( 0.572703 \)
Real period:	\( 7.996054 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 2 \)
Leading coefficient:	\( 4.579372 \)
Analytic order of Ш:	\( 4 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + T + 2 T^{2} )\)
\(3\)	\(2\)	\(2\)	\(1\)	\(1 + T^{2}\)
\(7\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 2 T + 7 T^{2} )\)
\(67\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 4 T + 67 T^{2} )\)
\(103\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 - 8 T + 103 T^{2} )\)

Galois representations

The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) except those listed.

Prime \(\ell\)	mod-\(\ell\) image	Is torsion prime?
\(2\)	2.15.1	yes

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$\mathrm{USp}(4)$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{USp}(4)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)	\(\simeq\)	\(\Z\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q\)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).