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⁡(JQ‾)⊗R\End(J_{\overline{\Q}}) \otimes \R: $\mathrm{M}_2(\R)$
• End⁡(JQ‾)⊗Q\End(J_{\overline{\Q}}) \otimes \Q: $\mathrm{M}_2(\Q)$
• End⁡(J)⊗Q\End(J) \otimes \Q: $\Q$
• Q‾\overline{\Q}-simple: no
• GL2\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