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

• Label: 2.625.ads_fgs
• 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:	$( 1 - 48 x + 625 x^{2} )^{2}$
	$1 - 96 x + 3554 x^{2} - 60000 x^{3} + 390625 x^{4}$
Frobenius angles:	$\pm0.0903344706017$, $\pm0.0903344706017$
Angle rank:	$1$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$334084$	$151766343184$	$59594591000161156$	$23282963759721716121600$	$9094946454304799222747208964$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$530$	$388518$	$244099442$	$152587231294$	$95367425732690$	$59604644903892198$	$37252902994479676082$	$23283064365779955110014$	$14551915228379552598511250$	$9094947017729646397023063078$

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.abw^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.625.a_aboo	$2$	(not in LMFDB)
2.625.ds_fgs	$2$	(not in LMFDB)
2.625.bw_cmp	$3$	(not in LMFDB)