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

• Label: 2.5.ab_ad
• 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:	$1 - x - 3 x^{2} - 5 x^{3} + 25 x^{4}$
Frobenius angles:	$\pm0.123439569429$, $\pm0.747770883447$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-26 +2 \sqrt{53}})\)
Galois group:	$D_{4}$
Jacobians:	$1$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$17$	$493$	$12869$	$422501$	$9656272$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$5$	$19$	$101$	$675$	$3090$	$15787$	$79049$	$390627$	$1957601$	$9770694$

Jacobians and polarizations

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

• $y^2=3 x^6+3 x^4+4 x^3+4 x^2+3 x+3$

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 \(\Q(\sqrt{-26 +2 \sqrt{53}})\).

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_ad	$2$	2.25.ah_bx