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

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

Invariants

Base field:	$\F_{5}$
Dimension:	$2$
L-polynomial:	$( 1 - 2 x + 5 x^{2} )( 1 - x + 5 x^{2} )$
	$1 - 3 x + 12 x^{2} - 15 x^{3} + 25 x^{4}$
Frobenius angles:	$\pm0.352416382350$, $\pm0.428216853436$
Angle rank:	$2$ (numerical)
Jacobians:	$0$
Isomorphism classes:	2
Cyclic group of points:	no
Non-cyclic primes:	$2$

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$20$	$1120$	$20720$	$380800$	$9208100$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$41$	$162$	$609$	$2943$	$15446$	$78627$	$391969$	$1953018$	$9761201$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 1.5.ac $\times$ 1.5.ab and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.5.ac : \(\Q(\sqrt{-1}) \).
• 1.5.ab : \(\Q(\sqrt{-19}) \).

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.ab_i	$2$	2.25.p_ea
2.5.b_i	$2$	2.25.p_ea
2.5.d_m	$2$	2.25.p_ea