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

• Label: 1.25.k
• Base field: $\F_{5^{2}}$
• Dimension: $1$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: yes
• Geometrically simple: yes
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{5^{2}}$
Dimension:	$1$
L-polynomial:	$( 1 + 5 x )^{2}$
	$1 + 10 x + 25 x^{2}$
Frobenius angles:	$1$, $1$
Angle rank:	$0$ (numerical)
Number field:	\(\Q\)
Galois group:	Trivial
Jacobians:	$1$

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

Newton polygon

This isogeny class is supersingular.

$p$-rank:	$0$
Slopes:	$[1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$36$	$576$	$15876$	$389376$	$9771876$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$36$	$576$	$15876$	$389376$	$9771876$	$244109376$	$6103671876$	$152587109376$	$3814701171876$	$95367412109376$

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 the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$.

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.a

Twists

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

Twist	Extension degree	Common base change
1.25.ak	$2$	1.625.aby
1.25.af	$3$	(not in LMFDB)
1.25.f	$6$	(not in LMFDB)