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

• Label: 2.5.ac_b
• 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:	$1 - 2 x + x^{2} - 10 x^{3} + 25 x^{4}$
Frobenius angles:	$\pm0.119187091912$, $\pm0.660634678170$
Angle rank:	$2$ (numerical)
Number field:	4.0.65600.5
Galois group:	$D_{4}$
Jacobians:	$2$
Isomorphism classes:	4
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$15$	$585$	$12060$	$403065$	$10080375$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$24$	$94$	$644$	$3224$	$15534$	$78824$	$392644$	$1953094$	$9772824$

Jacobians and polarizations

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

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

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 4.0.65600.5.

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_b	$2$	2.25.ac_l