Genus 2 curve 196.a.21952.1

• Label: 196.a.21952.1
• Conductor: $196$
• Discriminant: $-21952$
• Mordell-Weil group: \(\Z/{6}\Z \oplus \Z/{6}\Z\)
• Sato-Tate group: $E_1$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: no

This is a model for the modular curve $X_0(28)$.

Minimal equation

$y^2 + (x^2 + x)y = x^6 + 3x^5 + 6x^4 + 7x^3 + 6x^2 + 3x + 1$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(196\)	\(=\)	\( 2^{2} \cdot 7^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(-21952\)	\(=\)	\( - 2^{6} \cdot 7^{3} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(1340\)	\(=\)	\( 2^{2} \cdot 5 \cdot 67 \)
\( I_4 \)	\(=\)	\(1345\)	\(=\)	\( 5 \cdot 269 \)
\( I_6 \)	\(=\)	\(149855\)	\(=\)	\( 5 \cdot 17 \cdot 41 \cdot 43 \)
\( I_{10} \)	\(=\)	\(2809856\)	\(=\)	\( 2^{13} \cdot 7^{3} \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	$D_6$
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$D_6$

Rational points

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

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\((-1 : 1 : 1) - (1 : 1 : 0)\)	\(z (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3\)	\(0\)	\(6\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\)	\(3x^2 + 5xz + 4z^2\)	\(=\)	\(0,\)	\(9y\)	\(=\)	\(8xz^2 + 13z^3\)	\(0\)	\(6\)

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

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(2\)
Regulator:	\( 1 \)
Real period:	\( 11.77714 \)
Tamagawa product:	\( 12 \)
Torsion order:	\( 36 \)
Leading coefficient:	\( 0.109047 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

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

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?
\(2\)	2.360.3	yes
\(3\)	3.17280.1	yes

Sato-Tate group

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

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  Elliptic curve isogeny class 14.a

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)	\(\simeq\)	an Eichler order of index \(3\) in a maximal order of \(\End (J_{}) \otimes \Q\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\mathrm{M}_2 (\R)\)

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