Abelian variety isogeny class 2.25.au_fu over $\F_{5^{2}}$

• Label: 2.25.au_fu
• Base field: $\F_{5^{2}}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: no
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{5^{2}}$
Dimension:	$2$
L-polynomial:	$( 1 - 5 x )^{4}$
	$1 - 20 x + 150 x^{2} - 500 x^{3} + 625 x^{4}$
Frobenius angles:	$0$, $0$, $0$, $0$
Angle rank:	$0$ (numerical)
Jacobians:	$1$

This isogeny class is not 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:	$[1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$256$	$331776$	$236421376$	$151613669376$	$95245419909376$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$526$	$15126$	$388126$	$9753126$	$244078126$	$6103203126$	$152586328126$	$3814689453126$	$95367392578126$

Jacobians and polarizations

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

• $y^2=x^5+4x$

Decomposition and endomorphism algebra

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

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

The isogeny class factors as 1.25.ak^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ 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
$\F_{5}$	2.5.a_ak

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.25.a_aby	$2$	2.625.adw_fog
2.25.u_fu	$2$	2.625.adw_fog
2.25.af_a	$3$	(not in LMFDB)
2.25.k_cx	$3$	(not in LMFDB)