Genus 2 curve 6400.i.409600.1

• Label: 6400.i.409600.1
• Conductor: $6400$
• Discriminant: $-409600$
• Mordell-Weil group: \(\Z/{2}\Z\)
• Sato-Tate group: $E_1$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: no

Minimal equation

$y^2 = -x^6 - 4x^4 - 4x^2 - 1$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(6400\)	\(=\)	\( 2^{8} \cdot 5^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(-409600\)	\(=\)	\( - 2^{14} \cdot 5^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(248\)	\(=\)	\( 2^{3} \cdot 31 \)
\( I_4 \)	\(=\)	\(181\)	\(=\)	\( 181 \)
\( I_6 \)	\(=\)	\(14873\)	\(=\)	\( 107 \cdot 139 \)
\( I_{10} \)	\(=\)	\(50\)	\(=\)	\( 2 \cdot 5^{2} \)

Automorphism group

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

Rational points

This curve has no rational points.

Number of rational Weierstrass points: \(0\)

This curve is locally solvable except over $\R$, $\Q_{2}$, and $\Q_{3}$.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\(D_0 - D_\infty\)	\(x^2 + z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(2\)

2-torsion field: \(\Q(i, \sqrt{5})\)

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(1\)
Regulator:	\( 1 \)
Real period:	\( 5.171827 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 2 \)
Leading coefficient:	\( 1.292956 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(8\)	\(14\)	\(1\)	\(1\)
\(5\)	\(2\)	\(2\)	\(1\)	\(( 1 + T )^{2}\)

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?
\(2\)	2.180.4	yes
\(3\)	3.8640.12	no

Sato-Tate group

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

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  Elliptic curve isogeny class 80.b

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

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

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