Genus 2 curve 4150.a.41500.1

• Label: 4150.a.41500.1
• Conductor: $4150$
• Discriminant: $-41500$
• Mordell-Weil group: \(\Z \oplus \Z/{4}\Z\)
• 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

$y^2 + (x^2 + x + 1)y = x^6 + 3x^5 + 2x^4 - x - 1$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(4150\)	\(=\)	\( 2 \cdot 5^{2} \cdot 83 \)
Discriminant:	\( \Delta \)	\(=\)	\(-41500\)	\(=\)	\( - 2^{2} \cdot 5^{3} \cdot 83 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(756\)	\(=\)	\( 2^{2} \cdot 3^{3} \cdot 7 \)
\( I_4 \)	\(=\)	\(-35631\)	\(=\)	\( - 3^{2} \cdot 37 \cdot 107 \)
\( I_6 \)	\(=\)	\(-9830871\)	\(=\)	\( - 3^{2} \cdot 149 \cdot 7331 \)
\( I_{10} \)	\(=\)	\(-5312000\)	\(=\)	\( - 2^{9} \cdot 5^{3} \cdot 83 \)

Automorphism group

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

Rational points

All points
\((1 : -1 : 0)\)	\((1 : 1 : 0)\)	\((-1 : 0 : 1)\)	\((-1 : -1 : 1)\)	\((1 : 1 : 1)\)	\((1 : -4 : 1)\)
\((-3 : -7 : 2)\)	\((-13 : -475 : 8)\)	\((-13 : -557 : 8)\)

Number of rational Weierstrass points: \(1\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z \oplus \Z/{4}\Z\)

Generator	$D_0$						Height	Order
\((-3 : -7 : 2) + (-1 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\)	\((x + z) (2x + 3z)\)	\(=\)	\(0,\)	\(4y\)	\(=\)	\(7xz^2 + 7z^3\)	\(0.150550\)	\(\infty\)
\((-1 : 0 : 1) - (1 : -1 : 0)\)	\(z (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(x^3 + z^3\)	\(0\)	\(4\)

2-torsion field: 6.2.861125.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(1\)
Mordell-Weil rank:	\(1\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.150550 \)
Real period:	\( 14.51698 \)
Tamagawa product:	\( 4 \)
Torsion order:	\( 4 \)
Leading coefficient:	\( 0.546384 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(1\)	\(2\)	\(2\)	\(( 1 + T )( 1 - T + 2 T^{2} )\)
\(5\)	\(2\)	\(3\)	\(2\)	\(( 1 - T )( 1 + T )\)
\(83\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 83 T^{2} )\)

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?
\(2\)	2.60.1	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 \(\Q\).