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

• Label: 2.5.ac_f
• Base field: $\F_{5}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• 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:	$1 - 2 x + 5 x^{2} - 10 x^{3} + 25 x^{4}$
Frobenius angles:	$\pm0.219592142625$, $\pm0.605066734301$
Angle rank:	$2$ (numerical)
Number field:	4.0.83520.4
Galois group:	$D_{4}$
Jacobians:	$2$
Isomorphism classes:	2

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$19$	$817$	$14668$	$412585$	$10465219$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$32$	$118$	$660$	$3344$	$15662$	$77648$	$390820$	$1950814$	$9754352$

Jacobians and polarizations

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

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

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 4.0.83520.4.

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.c_f	$2$	2.25.g_bj