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

• Label: 2.5.b_a
• 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:	$1 + x + 5 x^{3} + 25 x^{4}$
Frobenius angles:	$\pm0.293537673823$, $\pm0.810347921095$
Angle rank:	$2$ (numerical)
Number field:	4.0.8405.1
Galois group:	$D_{4}$
Jacobians:	$4$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$32$	$640$	$17792$	$442880$	$9461152$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$7$	$25$	$142$	$705$	$3027$	$15670$	$77287$	$389985$	$1956022$	$9765825$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):

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

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 4.0.8405.1.

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.ab_a	$2$	2.25.ab_bo