Genus 2 curve 2299.b.25289.1

• Label: 2299.b.25289.1
• Conductor: $2299$
• Discriminant: $25289$
• Mordell-Weil group: \(\Z \oplus \Z/{15}\Z\)
• 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

$y^2 + y = x^6 - 3x^5 - x^4 + 7x^3 + x^2 - 5x - 2$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(2299\)	\(=\)	\( 11^{2} \cdot 19 \)
Discriminant:	\( \Delta \)	\(=\)	\(25289\)	\(=\)	\( 11^{3} \cdot 19 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(5320\)	\(=\)	\( 2^{3} \cdot 5 \cdot 7 \cdot 19 \)
\( I_4 \)	\(=\)	\(5908\)	\(=\)	\( 2^{2} \cdot 7 \cdot 211 \)
\( I_6 \)	\(=\)	\(10159984\)	\(=\)	\( 2^{4} \cdot 19^{2} \cdot 1759 \)
\( I_{10} \)	\(=\)	\(101156\)	\(=\)	\( 2^{2} \cdot 11^{3} \cdot 19 \)

Automorphism group

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

Rational points

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

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z \oplus \Z/{15}\Z\)

Generator	$D_0$						Height	Order
\(D_0 - 2 \cdot(1 : -1 : 0)\)	\(z^2\)	\(=\)	\(0,\)	\(2y\)	\(=\)	\(2x^3 - 3x^2z\)	\(1.330234\)	\(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\)	\(x^2 - 2z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(xz^2 - 2z^3\)	\(0\)	\(15\)

2-torsion field: 6.0.147136.2

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(1\)
Mordell-Weil rank:	\(1\)
2-Selmer rank:	\(1\)
Regulator:	\( 1.330234 \)
Real period:	\( 17.92309 \)
Tamagawa product:	\( 3 \)
Torsion order:	\( 15 \)
Leading coefficient:	\( 0.317892 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(11\)	\(2\)	\(3\)	\(3\)	\(( 1 - T )^{2}\)
\(19\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 19 T^{2} )\)

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?
\(2\)	2.15.2	no
\(3\)	3.720.4	yes

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 11.a
  Elliptic curve isogeny class 209.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 \(\Q\).