Genus 2 curve 578.a.2312.1

• Label: 578.a.2312.1
• Conductor: $578$
• Discriminant: $2312$
• Mordell-Weil group: \(\Z/{12}\Z\)
• 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

Minimal equation

$y^2 + (x^2 + x)y = x^5 - 2x^4 + 2x^3 - 2x^2 + x$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(578\)	\(=\)	\( 2 \cdot 17^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(2312\)	\(=\)	\( 2^{3} \cdot 17^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(228\)	\(=\)	\( 2^{2} \cdot 3 \cdot 19 \)
\( I_4 \)	\(=\)	\(705\)	\(=\)	\( 3 \cdot 5 \cdot 47 \)
\( I_6 \)	\(=\)	\(135777\)	\(=\)	\( 3 \cdot 45259 \)
\( I_{10} \)	\(=\)	\(295936\)	\(=\)	\( 2^{10} \cdot 17^{2} \)

Automorphism group

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

Rational points

All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (1 : 0 : 1),\, (1 : -2 : 1)\)

Number of rational Weierstrass points: \(2\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{12}\Z\)

Generator	$D_0$						Height	Order
\((0 : 0 : 1) + (1 : 0 : 1) - 2 \cdot(1 : 0 : 0)\)	\((-x + z) x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(12\)

2-torsion field: 4.0.1088.2

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(1\)
Regulator:	\( 1 \)
Real period:	\( 13.91029 \)
Tamagawa product:	\( 3 \)
Torsion order:	\( 12 \)
Leading coefficient:	\( 0.289797 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(1\)	\(3\)	\(3\)	\(( 1 - T )( 1 + T + 2 T^{2} )\)
\(17\)	\(2\)	\(2\)	\(1\)	\(( 1 - T )( 1 + T )\)

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 34.a
  Elliptic curve isogeny class 17.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 \(\Q\).