Genus 2 curve 55112.a.110224.1

• Label: 55112.a.110224.1
• Conductor: $55112$
• Discriminant: $-110224$
• Mordell-Weil group: \(\Z \oplus \Z \oplus \Z\)
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(55112\)	\(=\)	\( 2^{3} \cdot 83^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(-110224\)	\(=\)	\( - 2^{4} \cdot 83^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(360\)	\(=\)	\( 2^{3} \cdot 3^{2} \cdot 5 \)
\( I_4 \)	\(=\)	\(1032\)	\(=\)	\( 2^{3} \cdot 3 \cdot 43 \)
\( I_6 \)	\(=\)	\(31284\)	\(=\)	\( 2^{2} \cdot 3^{2} \cdot 11 \cdot 79 \)
\( I_{10} \)	\(=\)	\(440896\)	\(=\)	\( 2^{6} \cdot 83^{2} \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((0 : -1 : 1)\)	\((0 : 1 : 1)\)	\((-1 : -1 : 1)\)	\((1 : 1 : 1)\)
\((-1 : 3 : 1)\)	\((1 : -3 : 1)\)	\((-1 : -7 : 2)\)	\((1 : 7 : 2)\)	\((-1 : 12 : 2)\)	\((1 : -12 : 2)\)

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
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x^2 + z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(z^3\)	\(0.314145\)	\(\infty\)
\((1 : -3 : 1) - (1 : 0 : 0)\)	\(z (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3 - 2z^3\)	\(0.403604\)	\(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\)	\(z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0.354584\)	\(\infty\)

2-torsion field: 6.0.27556.1

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(3\)
Mordell-Weil rank:	\(3\)
2-Selmer rank:	\(3\)
Regulator:	\( 0.034279 \)
Real period:	\( 11.99160 \)
Tamagawa product:	\( 2 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.822140 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(3\)	\(4\)	\(2\)	\(1 + T + 2 T^{2}\)
\(83\)	\(2\)	\(2\)	\(1\)	\(( 1 + T )^{2}\)

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?
\(2\)	2.15.2	no
\(3\)	3.90.1	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

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 664.a
  Elliptic curve isogeny class 83.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 \(\Q\).