Abelian variety isogeny class 2.625.ado_ezd over $\F_{5^{4}}$

• Label: 2.625.ado_ezd
• Base field: $\F_{5^{4}}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{5^{4}}$
Dimension:	$2$
L-polynomial:	$( 1 - 49 x + 625 x^{2} )( 1 - 43 x + 625 x^{2} )$
	$1 - 92 x + 3357 x^{2} - 57500 x^{3} + 390625 x^{4}$
Frobenius angles:	$\pm0.0637685608585$, $\pm0.170463428383$
Angle rank:	$2$ (numerical)

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$336391$	$151905765825$	$59598625703581888$	$23283045886763785505625$	$9094947679704005460682452991$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$534$	$388876$	$244115970$	$152587769524$	$95367438581934$	$59604645086404126$	$37252902992598296670$	$23283064365523403605924$	$14551915228367790153420834$	$9094947017729247987397006876$

Jacobians and polarizations

This isogeny class contains a Jacobian, and hence is principally polarizable.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{4}}$.

Endomorphism algebra over $\F_{5^{4}}$

The isogeny class factors as 1.625.abx $\times$ 1.625.abr and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.625.abx : \(\Q(\sqrt{-11}) \).
• 1.625.abr : \(\Q(\sqrt{-651}) \).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.625.ag_abgz	$2$	(not in LMFDB)
2.625.g_abgz	$2$	(not in LMFDB)
2.625.do_ezd	$2$	(not in LMFDB)