Genus 2 curve 101833.a.101833.1

• Label: 101833.a.101833.1
• Conductor: $101833$
• Discriminant: $-101833$
• Mordell-Weil group: \(\Z \oplus \Z \oplus \Z\)
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(101833\)	\(=\)	\( 101833 \)
Discriminant:	\( \Delta \)	\(=\)	\(-101833\)	\(=\)	\( -101833 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(2892\)	\(=\)	\( 2^{2} \cdot 3 \cdot 241 \)
\( I_4 \)	\(=\)	\(140937\)	\(=\)	\( 3 \cdot 109 \cdot 431 \)
\( I_6 \)	\(=\)	\(148622235\)	\(=\)	\( 3 \cdot 5 \cdot 311 \cdot 31859 \)
\( I_{10} \)	\(=\)	\(13034624\)	\(=\)	\( 2^{7} \cdot 101833 \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((1 : 0 : 1)\)	\((0 : 2 : 1)\)	\((-2 : 1 : 1)\)	\((1 : -2 : 1)\)
\((2 : -2 : 1)\)	\((0 : -3 : 1)\)	\((-4 : 5 : 1)\)	\((-2 : 6 : 1)\)	\((2 : -7 : 1)\)	\((3 : -15 : 2)\)
\((3 : -20 : 2)\)	\((-4 : 58 : 1)\)	\((41 : -5696 : 40)\)	\((41 : -127225 : 40)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z \oplus \Z \oplus \Z\)

Generator	$D_0$						Height	Order
\((0 : -3 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(xz^2 - 3z^3\)	\(0.902176\)	\(\infty\)
\((1 : -2 : 1) - (1 : -1 : 0)\)	\(z (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-2z^3\)	\(0.494473\)	\(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x^2 - 2z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-2z^3\)	\(0.113953\)	\(\infty\)

2-torsion field: 6.0.6517312.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(3\)   (upper bound)
Mordell-Weil rank:	\(3\)
2-Selmer rank:	\(3\)
Regulator:	\( 0.047162 \)
Real period:	\( 19.00797 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.896464 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(101833\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 78 T + 101833 T^{2} )\)

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?
\(2\)	2.10.1	no

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 \(\Q\).