Genus 2 curve 4608.c.27648.1

• Label: 4608.c.27648.1
• Conductor: $4608$
• Discriminant: $-27648$
• 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

$y^2 = x^5 - x^4 + x^2 - x$

(, )

Invariants

Conductor:	$N$	$=$	$4608$	$=$	$2^{9} \cdot 3^{2}$
Discriminant:	$\Delta$	$=$	$-27648$	$=$	$- 2^{10} \cdot 3^{3}$

Igusa-Clebsch invariants

$I_2$	$=$	$24$	$=$	$2^{3} \cdot 3$
$I_4$	$=$	$-72$	$=$	$- 2^{3} \cdot 3^{2}$
$I_6$	$=$	$-180$	$=$	$- 2^{2} \cdot 3^{2} \cdot 5$
$I_{10}$	$=$	$108$	$=$	$2^{2} \cdot 3^{3}$

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
$(0 : 0 : 1) - (1 : 0 : 0)$	$x$	$=$	$0,$	$y$	$=$	$0$	$0$	$2$
$(-1 : 0 : 1) - (1 : 0 : 0)$	$x + z$	$=$	$0,$	$y$	$=$	$0$	$0$	$2$
$(-1 : 0 : 1) + (1 : 0 : 1) - 2 \cdot(1 : 0 : 0)$	$(x - z) (x + z)$	$=$	$0,$	$y$	$=$	$0$	$0$	$2$

2-torsion field: $\Q(\sqrt{-3})$

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	$0$
Mordell-Weil rank:	$0$
2-Selmer rank:	$3$
Regulator:	$1$
Real period:	$13.75615$
Tamagawa product:	$4$
Torsion order:	$8$
Leading coefficient:	$0.859759$
Analytic order of Ш:	$1$   (rounded)
Order of Ш:	square

Local invariants

Prime	ord($N$)	ord($\Delta$)	Tamagawa	Root number*	L-factor	Cluster picture	Tame reduction?
$2$	$9$	$10$	$2$	$1^*$	$1$		no
$3$	$2$	$3$	$2$	$1$	$1 + 3 T^{2}$		yes

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
$3$	3.540.5	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 $\Q (b) \simeq$ 4.0.432.1 with defining polynomial:
  $x^{4} - 3 x^{2} + 3$

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
  $g_4 = -480 b^{3} + 672 b^{2} + 576 b - 1008$
  $g_6 = -16704 b^{3} + 19008 b^{2} + 29952 b - 39744$
   Conductor norm: 1024
  $y^2 = x^3 - g_4 / 48 x - g_6 / 864$ with
  $g_4 = 480 b^{3} + 672 b^{2} - 576 b - 1008$
  $g_6 = 16704 b^{3} + 19008 b^{2} - 29952 b - 39744$
   Conductor norm: 1024

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 $K = \Q (a) \simeq$ 8.0.2985984.1 with defining polynomial $x^{8} - 2 x^{7} + 2 x^{6} - 2 x^{5} + 7 x^{4} - 10 x^{3} + 8 x^{2} - 4 x + 1$

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 $\frac{16}{11} a^{7} - 2 a^{6} + \frac{21}{11} a^{5} - \frac{23}{11} a^{4} + 9 a^{3} - \frac{105}{11} a^{2} + \frac{83}{11} a - \frac{30}{11}$ 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 $F \simeq$ $\Q(\sqrt{3})$ with generator $2 a^{7} - 4 a^{6} + 3 a^{5} - 3 a^{4} + 13 a^{3} - 19 a^{2} + 11 a - 4$ with minimal polynomial $x^{2} - 3$:

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

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

Over subfield $F \simeq$ $\Q(\sqrt{-3})$ with generator $\frac{18}{11} a^{7} - 2 a^{6} + \frac{14}{11} a^{5} - \frac{19}{11} a^{4} + 10 a^{3} - \frac{92}{11} a^{2} + \frac{48}{11} a - \frac{9}{11}$ with minimal polynomial $x^{2} - x + 1$:

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

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

Over subfield $F \simeq$ $\Q(\zeta_{12})$ with generator $\frac{3}{11} a^{7} - a^{6} + \frac{6}{11} a^{5} - \frac{5}{11} a^{4} + 2 a^{3} - \frac{52}{11} a^{2} + \frac{19}{11} a - \frac{7}{11}$ with minimal polynomial $x^{4} - x^{2} + 1$:

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

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

Over subfield $F \simeq$ 4.0.432.1 with generator $\frac{9}{11} a^{7} - 2 a^{6} + \frac{18}{11} a^{5} - \frac{15}{11} a^{4} + 6 a^{3} - \frac{112}{11} a^{2} + \frac{57}{11} a - \frac{21}{11}$ with minimal polynomial $x^{4} - 3 x^{2} + 3$:

$\End (J_{F})$	$\simeq$	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 $F \simeq$ 4.2.1728.1 with generator $-\frac{8}{11} a^{7} + a^{6} - \frac{5}{11} a^{5} + \frac{6}{11} a^{4} - 4 a^{3} + \frac{47}{11} a^{2} - \frac{14}{11} a + \frac{4}{11}$ with minimal polynomial $x^{4} - 2 x^{3} - 2 x + 1$:

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

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

Over subfield $F \simeq$ 4.2.1728.1 with generator $-a^{7} + a^{6} - a^{5} + a^{4} - 6 a^{3} + 4 a^{2} - 3 a + 1$ with minimal polynomial $x^{4} - 2 x^{3} - 2 x + 1$:

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

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

Over subfield $F \simeq$ 4.0.432.1 with generator $\frac{9}{11} a^{7} - a^{6} + \frac{7}{11} a^{5} - \frac{15}{11} a^{4} + 5 a^{3} - \frac{46}{11} a^{2} + \frac{24}{11} a - \frac{21}{11}$ with minimal polynomial $x^{4} - 3 x^{2} + 3$:

$\End (J_{F})$	$\simeq$	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