Genus 2 curve 537289.c.537289.1

• Label: 537289.c.537289.1
• Conductor: $537289$
• Discriminant: $-537289$
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(537289\)	\(=\)	\( 733^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(-537289\)	\(=\)	\( - 733^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(2884\)	\(=\)	\( 2^{2} \cdot 7 \cdot 103 \)
\( I_4 \)	\(=\)	\(572473\)	\(=\)	\( 11 \cdot 71 \cdot 733 \)
\( I_6 \)	\(=\)	\(406587037\)	\(=\)	\( 41 \cdot 83 \cdot 163 \cdot 733 \)
\( I_{10} \)	\(=\)	\(-68772992\)	\(=\)	\( - 2^{7} \cdot 733^{2} \)

Automorphism group

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

Rational points

Known points
\((1 : -1 : 0)\)	\((1 : 1 : 0)\)	\((0 : -1 : 1)\)	\((0 : 1 : 1)\)	\((1 : 0 : 1)\)	\((1 : -2 : 1)\)
\((-16 : 4441 : 15)\)	\((-16 : -4681 : 15)\)	\((15 : -6134 : 31)\)	\((31 : -7095 : 16)\)	\((15 : -15256 : 31)\)	\((31 : -16217 : 16)\)

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
\((1 : -2 : 1) - (1 : -1 : 0)\)	\(z (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(x^3 - 3z^3\)	\(0.509372\)	\(\infty\)
\((0 : 1 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\)	\(x (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-3xz^2 + z^3\)	\(0.509372\)	\(\infty\)

2-torsion field: 6.0.34386496.2

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.194595 \)
Real period:	\( 19.30717 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 3.757081 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(733\)	\(2\)	\(2\)	\(1\)	\(1 + 50 T + 733 T^{2}\)

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?
\(2\)	2.40.3	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 \(\Q (b)\) with defining polynomial:
  \(x^{6} - x^{5} - 305 x^{4} - 3190 x^{3} - 13345 x^{2} - 24521 x - 15791\)

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
  \(g_4 = -\frac{195527}{768} b^{5} + \frac{106703}{48} b^{4} + \frac{15517437}{256} b^{3} + \frac{262259057}{768} b^{2} + \frac{70172353}{96} b + \frac{392766647}{768}\)
  \(g_6 = \frac{148933139}{144} b^{5} - \frac{20301225907}{2304} b^{4} - \frac{191390106469}{768} b^{3} - \frac{410462090611}{288} b^{2} - \frac{2367815638657}{768} b - \frac{5002537483207}{2304}\)
   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 \(K = \Q (a)\) with defining polynomial \(x^{6} - x^{5} - 305 x^{4} - 3190 x^{3} - 13345 x^{2} - 24521 x - 15791\)

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 \(F \simeq \) \(\Q(\sqrt{733}) \) with generator \(\frac{1}{8} a^{5} - \frac{5}{8} a^{4} - \frac{71}{2} a^{3} - \frac{2063}{8} a^{2} - \frac{5333}{8} a - \frac{1073}{2}\) with minimal polynomial \(x^{2} - x - 183\):

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

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

Over subfield \(F \simeq \) 3.3.537289.1 with generator \(-\frac{1}{9} a^{5} + \frac{5}{9} a^{4} + \frac{95}{3} a^{3} + \frac{2050}{9} a^{2} + \frac{1718}{3} a + \frac{3905}{9}\) with minimal polynomial \(x^{3} - x^{2} - 244 x - 1276\):

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

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