Genus 2 curve 73728.d.884736.1

• Label: 73728.d.884736.1
• Conductor: $73728$
• Discriminant: $884736$
• Mordell-Weil group: \(\Z/{2}\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
• Sato-Tate group: $J(E_4)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\Q\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: no

Minimal equation

$y^2 = 2x^5 - 5x^3 + 3x$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(73728\)	\(=\)	\( 2^{13} \cdot 3^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(884736\)	\(=\)	\( 2^{15} \cdot 3^{3} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(195\)	\(=\)	\( 3 \cdot 5 \cdot 13 \)
\( I_4 \)	\(=\)	\(630\)	\(=\)	\( 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
\( I_6 \)	\(=\)	\(44910\)	\(=\)	\( 2 \cdot 3^{2} \cdot 5 \cdot 499 \)
\( I_{10} \)	\(=\)	\(108\)	\(=\)	\( 2^{2} \cdot 3^{3} \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_2$
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$D_4$

Rational points

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

Number of rational Weierstrass points: \(4\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\((0 : 0 : 1) + (1 : 0 : 1) - 2 \cdot(1 : 0 : 0)\)	\(x (x - z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\)	\(2x^2 - 3z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(2\)
\((-1 : 0 : 1) + (0 : 0 : 1) - 2 \cdot(1 : 0 : 0)\)	\(x (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(2\)

2-torsion field: \(\Q(\sqrt{6}) \)

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(3\)
Regulator:	\( 1 \)
Real period:	\( 11.16630 \)
Tamagawa product:	\( 8 \)
Torsion order:	\( 8 \)
Leading coefficient:	\( 1.395788 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(13\)	\(15\)	\(4\)	\(1\)
\(3\)	\(2\)	\(3\)	\(2\)	\(1 + 3 T^{2}\)

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?
\(2\)	2.360.2	yes
\(3\)	3.270.1	no

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$J(E_4)$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 4.2.55296.4 with defining polynomial:
  \(x^{4} - 24\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = 160 b^{2} + 480\)
  \(g_6 = -1792 b^{3} - 11520 b\)
   Conductor norm: 32
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = 160 b^{2} + 480\)
  \(g_6 = 1792 b^{3} + 11520 b\)
   Conductor norm: 32

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)	\(\simeq\)	\(\Z\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q\)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\R\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 8.0.12230590464.5 with defining polynomial \(x^{8} + 4 x^{6} + 18 x^{4} - 68 x^{2} + 49\)

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

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

\(\End (J_{\overline{\Q}})\)	\(\simeq\)	a non-Eichler order of index \(4\) 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 \(F \simeq \) \(\Q(\sqrt{-1}) \) with generator \(\frac{5}{56} a^{7} + \frac{27}{56} a^{5} + \frac{125}{56} a^{3} - \frac{193}{56} a\) with minimal polynomial \(x^{2} + 1\):

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

  Sato Tate group: $E_4$
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) \(\Q(\sqrt{-6}) \) with generator \(\frac{1}{4} a^{6} + \frac{5}{4} a^{4} + \frac{25}{4} a^{2} - \frac{39}{4}\) with minimal polynomial \(x^{2} + 6\):

\(\End (J_{F})\)	\(\simeq\)	\(\Z\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q\)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\R\)

  Sato Tate group: $J(E_2)$
  Not of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) \(\Q(\sqrt{6}) \) with generator \(\frac{1}{28} a^{7} + \frac{1}{7} a^{5} + \frac{11}{28} a^{3} - \frac{41}{14} a\) with minimal polynomial \(x^{2} - 6\):

\(\End (J_{F})\)	\(\simeq\)	\(\Z\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q\)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\R\)

  Sato Tate group: $J(E_2)$
  Not of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) \(\Q(i, \sqrt{6})\) with generator \(-\frac{1}{56} a^{7} + \frac{1}{8} a^{6} - \frac{1}{14} a^{5} + \frac{5}{8} a^{4} - \frac{11}{56} a^{3} + \frac{25}{8} a^{2} + \frac{41}{28} a - \frac{39}{8}\) with minimal polynomial \(x^{4} + 9\):

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

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

Over subfield \(F \simeq \) 4.0.55296.1 with generator \(\frac{1}{8} a^{6} + \frac{5}{8} a^{4} + \frac{21}{8} a^{2} - \frac{43}{8}\) with minimal polynomial \(x^{4} + 6\):

\(\End (J_{F})\)	\(\simeq\)	\(\Z [\sqrt{2}]\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{2}) \)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

  Sato Tate group: $J(E_1)$
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 4.0.55296.1 with generator \(\frac{5}{56} a^{7} + \frac{27}{56} a^{5} + \frac{125}{56} a^{3} - \frac{137}{56} a\) with minimal polynomial \(x^{4} + 6\):

\(\End (J_{F})\)	\(\simeq\)	\(\Z [\sqrt{2}]\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{2}) \)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

  Sato Tate group: $J(E_1)$
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 4.2.55296.4 with generator \(-\frac{5}{56} a^{7} + \frac{1}{8} a^{6} - \frac{27}{56} a^{5} + \frac{5}{8} a^{4} - \frac{125}{56} a^{3} + \frac{21}{8} a^{2} + \frac{137}{56} a - \frac{43}{8}\) with minimal polynomial \(x^{4} - 24\):

\(\End (J_{F})\)	\(\simeq\)	an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q\) \(\times\) \(\Q\)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

  Sato Tate group: $J(E_1)$
  Of \(\GL_2\)-type, not simple

Over subfield \(F \simeq \) 4.2.55296.4 with generator \(\frac{5}{56} a^{7} + \frac{1}{8} a^{6} + \frac{27}{56} a^{5} + \frac{5}{8} a^{4} + \frac{125}{56} a^{3} + \frac{21}{8} a^{2} - \frac{137}{56} a - \frac{43}{8}\) with minimal polynomial \(x^{4} - 24\):

\(\End (J_{F})\)	\(\simeq\)	an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q\) \(\times\) \(\Q\)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

  Sato Tate group: $J(E_1)$
  Of \(\GL_2\)-type, not simple