Abelian variety isogeny class 3.13.ai_x_abs over $\F_{13}$

• Label: 3.13.ai_x_abs
• Base field: $\F_{13}$
• 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:	$\F_{13}$
Dimension:	$3$
L-polynomial:	$1 - 8 x + 23 x^{2} - 44 x^{3} + 299 x^{4} - 1352 x^{5} + 2197 x^{6}$
Frobenius angles:	$\pm0.105271848666$, $\pm0.221891558171$, $\pm0.706075897016$
Angle rank:	$3$ (numerical)
Number field:	6.0.954288.1
Galois group:	$D_{6}$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1116$	$4379184$	$10170363564$	$23727539983104$	$51342352353201276$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$152$	$2106$	$29084$	$372426$	$4827560$	$62772198$	$815711804$	$10604405790$	$137859379592$

Jacobians and polarizations

This isogeny class contains a Jacobian, and hence is principally polarizable.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{13}$.

Endomorphism algebra over $\F_{13}$

The endomorphism algebra of this simple isogeny class is 6.0.954288.1.

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
3.13.i_x_bs	$2$	(not in LMFDB)
3.13.b_ak_abp	$3$	(not in LMFDB)
3.13.h_bs_gh	$3$	(not in LMFDB)