Genus 2 curve 24649.a.24649.1

• Label: 24649.a.24649.1
• Conductor: $24649$
• Discriminant: $24649$
• Mordell-Weil group: \(\Z/{2}\Z \oplus \Z/{2}\Z\)
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(24649\)	\(=\)	\( 157^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(24649\)	\(=\)	\( 157^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(676\)	\(=\)	\( 2^{2} \cdot 13^{2} \)
\( I_4 \)	\(=\)	\(17113\)	\(=\)	\( 109 \cdot 157 \)
\( I_6 \)	\(=\)	\(3248173\)	\(=\)	\( 17 \cdot 157 \cdot 1217 \)
\( I_{10} \)	\(=\)	\(3155072\)	\(=\)	\( 2^{7} \cdot 157^{2} \)

Automorphism group

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

Rational points

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

Number of rational Weierstrass points: \(3\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{2}\Z \oplus \Z/{2}\Z\)

Generator	$D_0$						Height	Order
\((-1 : 0 : 1) - (1 : 0 : 0)\)	\(x + z\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(2\)
\((0 : 0 : 1) - (1 : 0 : 0)\)	\(x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(2\)

2-torsion field: 3.3.24649.1

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(2\)
Regulator:	\( 1 \)
Real period:	\( 23.31292 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 4 \)
Leading coefficient:	\( 1.457058 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(157\)	\(2\)	\(2\)	\(1\)	\(1 - 14 T + 157 T^{2}\)

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?
\(2\)	2.240.1	yes
\(3\)	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 \(\Q (b) \simeq \) 6.6.95388992557.1 with defining polynomial:
  \(x^{6} - x^{5} - 65 x^{4} - 160 x^{3} + 20 x^{2} + 208 x + 64\)

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
  \(g_4 = -\frac{7769}{12288} b^{5} + \frac{59357}{12288} b^{4} + \frac{46087}{4096} b^{3} + \frac{9527}{3072} b^{2} - \frac{45709}{3072} b - \frac{643}{192}\)
  \(g_6 = \frac{5431415}{73728} b^{5} - \frac{69952763}{147456} b^{4} - \frac{35573845}{16384} b^{3} - \frac{43417879}{147456} b^{2} + \frac{11206189}{4096} b + \frac{33289495}{36864}\)
   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 \(K = \Q (a) \simeq \) 6.6.95388992557.1 with defining polynomial \(x^{6} - x^{5} - 65 x^{4} - 160 x^{3} + 20 x^{2} + 208 x + 64\)

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 \(F \simeq \) \(\Q(\sqrt{157}) \) with generator \(-\frac{1}{8} a^{5} + \frac{1}{4} a^{4} + \frac{31}{4} a^{3} + \frac{101}{8} a^{2} - \frac{35}{4} a - 10\) with minimal polynomial \(x^{2} - x - 39\):

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

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

Over subfield \(F \simeq \) 3.3.24649.1 with generator \(\frac{1}{18} a^{5} - \frac{7}{36} a^{4} - \frac{13}{4} a^{3} - \frac{23}{36} a^{2} + \frac{31}{3} a + \frac{20}{9}\) with minimal polynomial \(x^{3} - x^{2} - 52 x - 64\):

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

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