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

• Label: 1.625.j
• 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:	$1 + 9 x + 625 x^{2}$
Frobenius angles:	$\pm0.557609776697$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-2419}) \)
Galois group:	$C_2$
Jacobians:	$8$
Isomorphism classes:	8

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$635$	$391795$	$244124480$	$152587305315$	$95367446999675$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$635$	$391795$	$244124480$	$152587305315$	$95367446999675$	$59604645002978560$	$37252902972971443595$	$23283064365349549705155$	$14551915228374468335954240$	$9094947017729237213627487475$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which 0 are hyperelliptic):

• $y^2=x^3+a^{32} x+a^{32}$
• $y^2=x^3+a^{256} x+a^{256}$
• $y^2=x^3+a^{496} x+a^{496}$
• $y^2=x^3+a^{224} x+a^{224}$
• $y^2=x^3+a^{608} x+a^{608}$
• $y^2=x^3+a^{160} x+a^{160}$
• $y^2=x^3+a^{176} x+a^{176}$
• $y^2=x^3+a^{544} x+a^{544}$

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 \(\Q(\sqrt{-2419}) \).

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.aj	$2$	(not in LMFDB)