Genus 2 curve 727609.a.727609.1

• Label: 727609.a.727609.1
• Conductor: $727609$
• Discriminant: $-727609$
• 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

$y^2 + (x^3 + x + 1)y = -3x^5 + 11x^4 - 11x^3 + x$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(727609\)	\(=\)	\( 853^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(-727609\)	\(=\)	\( - 853^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(3364\)	\(=\)	\( 2^{2} \cdot 29^{2} \)
\( I_4 \)	\(=\)	\(768553\)	\(=\)	\( 17 \cdot 53 \cdot 853 \)
\( I_6 \)	\(=\)	\(637949317\)	\(=\)	\( 853 \cdot 747889 \)
\( I_{10} \)	\(=\)	\(-93133952\)	\(=\)	\( - 2^{7} \cdot 853^{2} \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((0 : 0 : 1)\)	\((0 : -1 : 1)\)	\((1 : -1 : 1)\)	\((1 : -2 : 1)\)
\((-1 : -3 : 4)\)	\((-1 : -44 : 4)\)	\((5 : -45 : 1)\)	\((5 : -86 : 1)\)	\((4 : -124 : 5)\)	\((4 : -165 : 5)\)

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
\((0 : -1 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0.255966\)	\(\infty\)
\((1 : -1 : 1) - (1 : 0 : 0)\)	\(z (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3\)	\(0.255966\)	\(\infty\)

2-torsion field: 6.0.46566976.1

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.049139 \)
Real period:	\( 18.72177 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.919970 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(853\)	\(2\)	\(2\)	\(1\)	\(1 - 35 T + 853 T^{2}\)

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:
  \(x^{6} - x^{5} - 355 x^{4} + 2164 x^{3} + 12951 x^{2} - 127278 x + 250479\)

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
  \(g_4 = -\frac{91753778777}{1310256} b^{5} - \frac{427172560315}{1310256} b^{4} + \frac{15100223464969}{655128} b^{3} - \frac{26896534681157}{1310256} b^{2} - \frac{223789793030201}{218376} b + \frac{224889490077251}{72792}\)
  \(g_6 = \frac{3723001140950777}{5241024} b^{5} + \frac{16647437391232375}{5241024} b^{4} - \frac{309666823897929449}{1310256} b^{3} + \frac{1131323133896869361}{5241024} b^{2} + \frac{9177326539137560825}{873504} b - \frac{18418535473094813119}{582336}\)
   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 \(x^{6} - x^{5} - 355 x^{4} + 2164 x^{3} + 12951 x^{2} - 127278 x + 250479\)

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 \(F \simeq \) \(\Q(\sqrt{853}) \) with generator \(\frac{10}{81891} a^{5} + \frac{41}{81891} a^{4} - \frac{2431}{81891} a^{3} + \frac{17431}{81891} a^{2} - \frac{14851}{27297} a - \frac{124508}{9099}\) with minimal polynomial \(x^{2} - x - 213\):

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

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

Over subfield \(F \simeq \) 3.3.727609.1 with generator \(-\frac{100}{9099} a^{5} - \frac{410}{9099} a^{4} + \frac{33409}{9099} a^{3} - \frac{46924}{9099} a^{2} - \frac{512684}{3033} a + \frac{564677}{1011}\) with minimal polynomial \(x^{3} - x^{2} - 284 x - 1011\):

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

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