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

• Label: 2.5.ae_o
• 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:	$( 1 - 2 x + 5 x^{2} )^{2}$
	$1 - 4 x + 14 x^{2} - 20 x^{3} + 25 x^{4}$
Frobenius angles:	$\pm0.352416382350$, $\pm0.352416382350$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$16$	$1024$	$21904$	$409600$	$9265936$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$38$	$170$	$654$	$2962$	$15158$	$78010$	$392734$	$1957922$	$9764678$

Jacobians and polarizations

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

• $y^2=3x^6+3x^4+3x^2+3$

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^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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_g	$2$	2.25.m_di
2.5.e_o	$2$	2.25.m_di
2.5.c_ab	$3$	2.125.bs_bcg