Abelian variety isogeny class 3.5.a_m_b over $\F_{5}$

• Label: 3.5.a_m_b
• Base field: $\F_{5}$
• Dimension: $3$
• $p$-rank: $3$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{5}$
Dimension:	$3$
L-polynomial:	$1 + 12 x^{2} + x^{3} + 60 x^{4} + 125 x^{6}$
Frobenius angles:	$\pm0.388697363426$, $\pm0.475255832720$, $\pm0.638055277619$
Angle rank:	$3$ (numerical)
Number field:	6.0.37916019.1
Galois group:	$A_4\times C_2$
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:	$3$
Slopes:	$[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$199$	$39203$	$1991791$	$226475731$	$29931546419$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$50$	$129$	$578$	$3066$	$15509$	$78714$	$391970$	$1949412$	$9761930$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is not hyperelliptic):

• $x^4+x^3 y+3 x^3 z+2 x^2 y^2+3 x^2 y z+x^2 z^2+x z^3+y^3 z=0$

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 6.0.37916019.1.

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
3.5.a_m_ab	$2$	3.25.y_ke_cmz