Genus 2 curve 778752.c.778752.1

• Label: 778752.c.778752.1
• Conductor: $778752$
• Discriminant: $-778752$
• Mordell-Weil group: \(\Z \oplus \Z \oplus \Z/{2}\Z\)
• Sato-Tate group: $J(E_1)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\Q \times \Q\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Minimal equation

$y^2 + y = -6x^6 - 10x^4 - 5x^2 - 1$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(778752\)	\(=\)	\( 2^{9} \cdot 3^{2} \cdot 13^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(-778752\)	\(=\)	\( - 2^{9} \cdot 3^{2} \cdot 13^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(1880\)	\(=\)	\( 2^{3} \cdot 5 \cdot 47 \)
\( I_4 \)	\(=\)	\(1405\)	\(=\)	\( 5 \cdot 281 \)
\( I_6 \)	\(=\)	\(879765\)	\(=\)	\( 3 \cdot 5 \cdot 89 \cdot 659 \)
\( I_{10} \)	\(=\)	\(3042\)	\(=\)	\( 2 \cdot 3^{2} \cdot 13^{2} \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_2^2$
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$D_4$

Rational points

This curve has no rational points.

Number of rational Weierstrass points: \(0\)

This curve is locally solvable except over $\R$.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\(D_0 - D_\infty\)	\(5x^2 + 3z^2\)	\(=\)	\(0,\)	\(10y\)	\(=\)	\(3xz^2 - 5z^3\)	\(2.675228\)	\(\infty\)
\(D_0 - D_\infty\)	\(x^2 + z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0.762168\)	\(\infty\)
\(D_0 - D_\infty\)	\(2x^2 + z^2\)	\(=\)	\(0,\)	\(2y\)	\(=\)	\(-z^3\)	\(0\)	\(2\)

2-torsion field: 8.0.151613669376.4

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(4\)
Regulator:	\( 2.038973 \)
Real period:	\( 4.813692 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 2 \)
Leading coefficient:	\( 4.907496 \)
Analytic order of Ш:	\( 2 \)   (rounded)
Order of Ш:	twice a square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(9\)	\(9\)	\(1\)	\(1\)
\(3\)	\(2\)	\(2\)	\(1\)	\(( 1 - T )( 1 + T )\)
\(13\)	\(2\)	\(2\)	\(1\)	\(( 1 - T )( 1 + T )\)

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?
\(2\)	2.90.5	yes
\(3\)	3.1080.10	no

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$J(E_1)$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{SU}(2)\)

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 2496.u
  Elliptic curve isogeny class 312.b

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)	\(\simeq\)	an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{2}) \) with defining polynomial \(x^{2} - 2\)

Not of \(\GL_2\)-type over \(\overline{\Q}\)

Endomorphism ring over \(\overline{\Q}\):

\(\End (J_{\overline{\Q}})\)	\(\simeq\)	a non-Eichler order of index \(4\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)	\(\simeq\)	\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)	\(\simeq\)	\(\mathrm{M}_2 (\R)\)