Abelian variety isogeny class 1.25.j over $\F_{5^{2}}$

• Label: 1.25.j
• Base field: $\F_{5^{2}}$
• Dimension: $1$
• $p$-rank: $1$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{5^{2}}$
Dimension:	$1$
L-polynomial:	$1 + 9 x + 25 x^{2}$
Frobenius angles:	$\pm0.856433706871$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-19}) \)
Galois group:	$C_2$
Jacobians:	$1$
Isomorphism classes:	1

This isogeny class is simple and geometrically simple, not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$35$	$595$	$15680$	$390915$	$9761675$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$35$	$595$	$15680$	$390915$	$9761675$	$244168960$	$6103359395$	$152588588355$	$3814694891840$	$95367435561475$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{5^{2}}$.

Endomorphism algebra over $\F_{5^{2}}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-19}) \).

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{5^{2}}$.

Subfield	Primitive Model
$\F_{5}$	1.5.ab
$\F_{5}$	1.5.b

Twists

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

Twist	Extension degree	Common base change
1.25.aj	$2$	1.625.abf