Genus 2 curve 110889.a.110889.1

• Label: 110889.a.110889.1
• Conductor: $110889$
• Discriminant: $110889$
• Mordell-Weil group: \(\Z \oplus \Z\)
• Sato-Tate group: $E_6$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\mathsf{CM}\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Minimal equation

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(110889\)	\(=\)	\( 3^{4} \cdot 37^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(110889\)	\(=\)	\( 3^{4} \cdot 37^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(1380\)	\(=\)	\( 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
\( I_4 \)	\(=\)	\(94905\)	\(=\)	\( 3^{3} \cdot 5 \cdot 19 \cdot 37 \)
\( I_6 \)	\(=\)	\(34200765\)	\(=\)	\( 3^{3} \cdot 5 \cdot 37 \cdot 41 \cdot 167 \)
\( I_{10} \)	\(=\)	\(14193792\)	\(=\)	\( 2^{7} \cdot 3^{4} \cdot 37^{2} \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_6$
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$D_6$

Rational points

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

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z \oplus \Z\)

Generator	$D_0$						Height	Order
\((0 : -1 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0.228168\)	\(\infty\)
\((1 : -1 : 1) - (1 : 0 : 0)\)	\(z (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3\)	\(0.228168\)	\(\infty\)

2-torsion field: 9.9.1363532208525369.2

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.039045 \)
Real period:	\( 21.98056 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.858248 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(3\)	\(4\)	\(4\)	\(1\)	\(1 + 3 T + 3 T^{2}\)
\(37\)	\(2\)	\(2\)	\(1\)	\(1 + 10 T + 37 T^{2}\)

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?
\(2\)	2.80.1	no
\(3\)	3.480.12	no

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$E_6$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 6.6.454965701877.2 with defining polynomial:
  \(x^{6} - 111 x^{4} - 296 x^{3} + 999 x^{2} + 3663 x + 2331\)

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = -\frac{69588900}{7686973} b^{5} + \frac{195589215}{7686973} b^{4} + \frac{4154341500}{7686973} b^{3} + \frac{23045260005}{7686973} b^{2} + \frac{53213628105}{7686973} b + \frac{5891307795}{1098139}\)
  \(g_6 = \frac{7126269800796}{2636631739} b^{5} - \frac{78121867595337}{2636631739} b^{4} - \frac{220022576402778}{2636631739} b^{3} + \frac{1682832880289313}{2636631739} b^{2} + \frac{5514370690925493}{2636631739} b + \frac{502485786075150}{376661677}\)
   Conductor norm: 1

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)	\(\simeq\)	\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-3}) \)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\C\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 6.6.454965701877.2 with defining polynomial \(x^{6} - 111 x^{4} - 296 x^{3} + 999 x^{2} + 3663 x + 2331\)

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

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

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

Remainder of the endomorphism lattice by field

Over subfield \(F \simeq \) \(\Q(\sqrt{37}) \) with generator \(-\frac{225}{22411} a^{5} + \frac{690}{22411} a^{4} + \frac{22859}{22411} a^{3} - \frac{4995}{22411} a^{2} - \frac{209457}{22411} a - \frac{87714}{22411}\) with minimal polynomial \(x^{2} - x - 9\):

\(\End (J_{F})\)	\(\simeq\)	\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\C\)

  Sato Tate group: $E_3$
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 3.3.110889.1 with generator \(-\frac{8}{921} a^{5} + \frac{15}{307} a^{4} + \frac{250}{307} a^{3} - \frac{2081}{921} a^{2} - \frac{2331}{307} a + \frac{1924}{307}\) with minimal polynomial \(x^{3} - 111 x - 37\):

\(\End (J_{F})\)	\(\simeq\)	\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\C\)

  Sato Tate group: $E_2$
  Of \(\GL_2\)-type, simple