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

• Label: 2.5.ac_l
• 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: yes

Invariants

Base field:	$\F_{5}$
Dimension:	$2$
L-polynomial:	$( 1 - x + 5 x^{2} )^{2}$
	$1 - 2 x + 11 x^{2} - 10 x^{3} + 25 x^{4}$
Frobenius angles:	$\pm0.428216853436$, $\pm0.428216853436$
Angle rank:	$1$ (numerical)
Jacobians:	$1$

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$25$	$1225$	$19600$	$354025$	$9150625$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$44$	$154$	$564$	$2924$	$15734$	$79244$	$391204$	$1948114$	$9757724$

Jacobians and polarizations

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

• $y^2=4x^6+x^5+x^4+x^3+x^2+x+4$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{5}$

The isogeny class factors as 1.5.ab^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-19}) \)$)$

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.a_j	$2$	2.25.s_fb
2.5.c_l	$2$	2.25.s_fb
2.5.b_ae	$3$	2.125.bc_re