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

• Label: 1.625.ad
• 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:	$1 - 3 x + 625 x^{2}$
Frobenius angles:	$\pm0.480889929069$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-2491}) \)
Galois group:	$C_2$
Jacobians:	$12$
Isomorphism classes:	12

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$623$	$391867$	$244146224$	$152587131795$	$95367425865383$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$623$	$391867$	$244146224$	$152587131795$	$95367425865383$	$59604645232334272$	$37252902989599498439$	$23283064365116314185315$	$14551915228362927136891568$	$9094947017729439760581962827$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which 0 are hyperelliptic):

• $y^2=x^3+a^{19} x+a^{19}$
• $y^2=x^3+a^{339} x+a^{340}$
• $y^2=x^3+a^{499} x+a^{500}$
• $y^2=x^3+a^{475} x+a^{475}$
• $y^2=x^3+a^{95} x+a^{95}$
• $y^2=x^3+a^{319} x+a^{320}$
• $y^2=x^3+a^{571} x+a^{572}$
• $y^2=x^3+a^{315} x+a^{316}$
• $y^2=x^3+a^{615} x+a^{616}$
• $y^2=x^3+a^{439} x+a^{440}$
• $y^2=x^3+a^{503} x+a^{503}$
• $y^2=x^3+a^{351} x+a^{352}$

where $a$ is a root of the Conway polynomial.

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 \(\Q(\sqrt{-2491}) \).

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.d	$2$	(not in LMFDB)