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

• Label: 3.13.aq_el_atn
• Base field: $\F_{13}$
• Dimension: $3$
• $p$-rank: $2$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{13}$
Dimension:	$3$
L-polynomial:	$( 1 - 7 x + 13 x^{2} )( 1 - 9 x + 39 x^{2} - 117 x^{3} + 169 x^{4} )$
	$1 - 16 x + 115 x^{2} - 507 x^{3} + 1495 x^{4} - 2704 x^{5} + 2197 x^{6}$
Frobenius angles:	$\pm0.0228181011636$, $\pm0.0772104791556$, $\pm0.419357734967$
Angle rank:	$3$ (numerical)

This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.

Newton polygon

$p$-rank:	$2$
Slopes:	$[0, 0, 1/2, 1/2, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$581$	$4087335$	$10141926704$	$22738764255375$	$50810670439501616$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$144$	$2101$	$27868$	$368563$	$4822533$	$62752156$	$815728532$	$10604296903$	$137857882839$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{13}$

The isogeny class factors as 1.13.ah $\times$ 2.13.aj_bn and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.13.ah : \(\Q(\sqrt{-3}) \).
• 2.13.aj_bn : 4.0.10933.1.

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.ac_al_bn	$2$	(not in LMFDB)
3.13.c_al_abn	$2$	(not in LMFDB)
3.13.q_el_tn	$2$	(not in LMFDB)
3.13.ah_bi_aga	$3$	(not in LMFDB)
3.13.ae_h_abn	$3$	(not in LMFDB)