Abelian variety isogeny class 1.625.ae over $\F_{5^{4}}$

• Label: 1.625.ae
• Base field: $\F_{5^{4}}$
• Dimension: $1$
• $p$-rank: $1$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{5^{4}}$
Dimension:	$1$
L-polynomial:	$1 - 4 x + 625 x^{2}$
Frobenius angles:	$\pm0.474507968135$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-69}) \)
Galois group:	$C_2$

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$622$	$391860$	$244148062$	$152587149120$	$95367424027102$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$622$	$391860$	$244148062$	$152587149120$	$95367424027102$	$59604645208377780$	$37252902991109541742$	$23283064365142307523840$	$14551915228361816684475982$	$9094947017729415148265973300$

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 endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-69}) \).

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
1.625.e	$2$	(not in LMFDB)