Genus 2 curve 893.a.893.1

• Label: 893.a.893.1
• Conductor: $893$
• Discriminant: $893$
• Mordell-Weil group: $\Z$
• Sato-Tate group: $\mathrm{USp}(4)$
• End⁡(JQ‾)⊗R\End(J_{\overline{\Q}}) \otimes \R: $\R$
• End⁡(JQ‾)⊗Q\End(J_{\overline{\Q}}) \otimes \Q: $\Q$
• End⁡(J)⊗Q\End(J) \otimes \Q: $\Q$
• Q‾\overline{\Q}-simple: yes
• GL2\mathrm{GL}_2-type: no

Minimal equation

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

(, )

Invariants

Conductor:	$N$	$=$	$893$	$=$	$19 \cdot 47$
Discriminant:	$\Delta$	$=$	$893$	$=$	$19 \cdot 47$

Igusa-Clebsch invariants

$I_2$	$=$	$156$	$=$	$2^{2} \cdot 3 \cdot 13$
$I_4$	$=$	$-519$	$=$	$- 3 \cdot 173$
$I_6$	$=$	$-11805$	$=$	$- 3 \cdot 5 \cdot 787$
$I_{10}$	$=$	$-114304$	$=$	$- 2^{7} \cdot 19 \cdot 47$

Automorphism group

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

Rational points

All points
$(1 : 0 : 0)$	$(1 : -1 : 0)$	$(0 : 0 : 1)$	$(0 : -1 : 1)$	$(1 : -1 : 1)$	$(1 : -2 : 1)$
$(-2 : 4 : 1)$	$(-2 : 5 : 1)$

Number of rational Weierstrass points: $0$

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: $\Z$

Generator	$D_0$						Height	Order
$D_0 - 2 \cdot(1 : -1 : 0)$	$z^2$	$=$	$0,$	$y$	$=$	$0$	$0.006429$	$\infty$

2-torsion field: 6.2.57152.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	$1$
Mordell-Weil rank:	$1$
2-Selmer rank:	$1$
Regulator:	$0.006429$
Real period:	$23.40243$
Tamagawa product:	$1$
Torsion order:	$1$
Leading coefficient:	$0.150458$
Analytic order of Ш:	$1$   (rounded)
Order of Ш:	square

Local invariants

Prime	ord($N$)	ord($\Delta$)	Tamagawa	L-factor	Cluster picture
$19$	$1$	$1$	$1$	$( 1 - T )( 1 + 5 T + 19 T^{2} )$
$47$	$1$	$1$	$1$	$( 1 + T )( 1 + 3 T + 47 T^{2} )$

Galois representations

The mod-$\ell$ Galois representation has maximal image $\GSp(4,\F_\ell)$ for all primes $\ell$ .

Sato-Tate group

$\mathrm{ST}$	$\simeq$	$\mathrm{USp}(4)$
$\mathrm{ST}^0$	$\simeq$	$\mathrm{USp}(4)$

Decomposition of the Jacobian

Simple over $\overline{\Q}$

Endomorphisms of the Jacobian

Not of $\GL_2$-type over $\Q$

Endomorphism ring over $\Q$:

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

All $\overline{\Q}$-endomorphisms of the Jacobian are defined over $\Q$.