Genus 2 curve 21609.a.453789.1

• Label: 21609.a.453789.1
• Conductor: $21609$
• Discriminant: $-453789$
• Mordell-Weil group: \(\Z \oplus \Z \oplus \Z/{2}\Z\)
• Sato-Tate group: $\mathrm{SU}(2)\times\mathrm{SU}(2)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R \times \R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathsf{RM}\)
• \(\End(J) \otimes \Q\): \(\mathsf{RM}\)
• \(\overline{\Q}\)-simple: yes
• \(\mathrm{GL}_2\)-type: yes

Minimal equation

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(21609\)	\(=\)	\( 3^{2} \cdot 7^{4} \)
Discriminant:	\( \Delta \)	\(=\)	\(-453789\)	\(=\)	\( - 3^{3} \cdot 7^{5} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(116\)	\(=\)	\( 2^{2} \cdot 29 \)
\( I_4 \)	\(=\)	\(793\)	\(=\)	\( 13 \cdot 61 \)
\( I_6 \)	\(=\)	\(25821\)	\(=\)	\( 3^{2} \cdot 19 \cdot 151 \)
\( I_{10} \)	\(=\)	\(3456\)	\(=\)	\( 2^{7} \cdot 3^{3} \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((0 : 1 : 1)\)	\((1 : 0 : 1)\)	\((-2 : 0 : 1)\)	\((0 : -2 : 1)\)
\((1 : -2 : 1)\)	\((-2 : 7 : 1)\)	\((3 : -7 : 2)\)	\((3 : -28 : 2)\)	\((-5 : -28 : 3)\)	\((-5 : 126 : 3)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\(D_0 - 2 \cdot(1 : -1 : 0)\)	\(z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0.304655\)	\(\infty\)
\((0 : -2 : 1) - (1 : 0 : 0)\)	\(z x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3 - 2z^3\)	\(0.152327\)	\(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x^2 - xz + z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(2\)

2-torsion field: 8.0.152473104.1

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(3\)
Regulator:	\( 0.045400 \)
Real period:	\( 13.61578 \)
Tamagawa product:	\( 4 \)
Torsion order:	\( 2 \)
Leading coefficient:	\( 0.618161 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(3\)	\(2\)	\(3\)	\(2\)	\(( 1 + T )^{2}\)
\(7\)	\(4\)	\(5\)	\(2\)	\(1\)

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?
\(2\)	2.45.1	yes
\(3\)	3.72.2	no

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

Simple over \(\overline{\Q}\)

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)	\(\simeq\)	\(\Z [\sqrt{2}]\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{2}) \)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

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