Genus 2 curve 3969.d.250047.1

• Label: 3969.d.250047.1
• Conductor: $3969$
• Discriminant: $-250047$
• Mordell-Weil group: \(\Z/{2}\Z \oplus \Z/{6}\Z\)
• Sato-Tate group: $J(E_1)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\mathsf{RM}\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Minimal equation

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(3969\)	\(=\)	\( 3^{4} \cdot 7^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(-250047\)	\(=\)	\( - 3^{6} \cdot 7^{3} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(452\)	\(=\)	\( 2^{2} \cdot 113 \)
\( I_4 \)	\(=\)	\(-15543\)	\(=\)	\( - 3^{2} \cdot 11 \cdot 157 \)
\( I_6 \)	\(=\)	\(-660459\)	\(=\)	\( - 3 \cdot 19 \cdot 11587 \)
\( I_{10} \)	\(=\)	\(131712\)	\(=\)	\( 2^{7} \cdot 3 \cdot 7^{3} \)

Automorphism group

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

Rational points

All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (-1 : -26 : 4)\)

Number of rational Weierstrass points: \(2\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{2}\Z \oplus \Z/{6}\Z\)

Generator	$D_0$						Height	Order
\(D_0 - 2 \cdot(1 : 0 : 0)\)	\(3x^2 - 3xz - z^2\)	\(=\)	\(0,\)	\(3y\)	\(=\)	\(-3xz^2 - 2z^3\)	\(0\)	\(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\)	\(3x^2 - 2xz - 2z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-xz^2\)	\(0\)	\(6\)

2-torsion field: \(\Q(\sqrt{-3}, \sqrt{-7})\)

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(2\)
Regulator:	\( 1 \)
Real period:	\( 13.55904 \)
Tamagawa product:	\( 8 \)
Torsion order:	\( 12 \)
Leading coefficient:	\( 0.753280 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(3\)	\(4\)	\(6\)	\(4\)	\(1\)
\(7\)	\(2\)	\(3\)	\(2\)	\(( 1 - T )^{2}\)

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?
\(2\)	2.180.3	yes
\(3\)	3.1920.1	yes

Sato-Tate group

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

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\sqrt{-3}) \) with defining polynomial:
  \(x^{2} - x + 1\)

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  Elliptic curve isogeny class 2.0.3.1-441.2-a

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)	\(\simeq\)	\(\Z [\sqrt{3}]\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{3}) \)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{-3}) \) with defining polynomial \(x^{2} - x + 1\)

Not of \(\GL_2\)-type over \(\overline{\Q}\)

Endomorphism ring over \(\overline{\Q}\):

\(\End (J_{\overline{\Q}})\)	\(\simeq\)	a non-Eichler order of index \(18\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)	\(\simeq\)	\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)	\(\simeq\)	\(\mathrm{M}_2 (\R)\)