Abelian variety isogeny class 2.5.a_k over $\F_{5}$

• Label: 2.5.a_k
• Base field: $\F_{5}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{5}$
Dimension:	$2$
L-polynomial:	$( 1 + 5 x^{2} )^{2}$
	$1 + 10 x^{2} + 25 x^{4}$
Frobenius angles:	$\pm0.5$, $\pm0.5$
Angle rank:	$0$ (numerical)
Jacobians:	$2$

This isogeny class is not simple, 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$36$	$1296$	$15876$	$331776$	$9771876$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$46$	$126$	$526$	$3126$	$16126$	$78126$	$388126$	$1953126$	$9778126$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):

• $y^2=x^5+x$
• $y^2=x^6+3$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 1.5.a^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-5}) \)$)$

Endomorphism algebra over $\overline{\F}_{5}$

The base change of $A$ to $\F_{5^{2}}$ is 1.25.k^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 is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.5.a_af	$3$	2.125.a_jq
2.5.a_ak	$4$	2.625.adw_fog
2.5.a_a	$8$	(not in LMFDB)