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

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

Invariants

Base field:	$\F_{5}$
Dimension:	$2$
L-polynomial:	$1 - x - x^{2} - 5 x^{3} + 25 x^{4}$
Frobenius angles:	$\pm0.169335136474$, $\pm0.720319309201$
Angle rank:	$2$ (numerical)
Number field:	4.0.1525.1
Galois group:	$D_{4}$
Jacobians:	$3$
Isomorphism classes:	3

This isogeny class is simple and geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$19$	$589$	$13471$	$438805$	$9968464$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$5$	$23$	$107$	$699$	$3190$	$15743$	$79147$	$390259$	$1952765$	$9767078$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:

• $y^2=4x^6+3x^5+x^4+4x^3+3x^2+4x+3$
• $y^2=4x^6+4x^4+4x^3+3x^2+2x+2$
• $y^2=2x^6+2x^5+4x^4+3x^2+3$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{5}$

The endomorphism algebra of this simple isogeny class is 4.0.1525.1.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.5.b_ab	$2$	2.25.ad_bp