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

• Label: 3.13.aq_es_avi
• Base field: $\F_{13}$
• Dimension: $3$
• $p$-rank: $3$
• Ordinary: yes
• 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:	$( 1 - 6 x + 13 x^{2} )( 1 - 10 x + 49 x^{2} - 130 x^{3} + 169 x^{4} )$
	$1 - 16 x + 122 x^{2} - 554 x^{3} + 1586 x^{4} - 2704 x^{5} + 2197 x^{6}$
Frobenius angles:	$\pm0.151058869957$, $\pm0.187167041811$, $\pm0.334339837461$
Angle rank:	$3$ (numerical)

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

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$632$	$4537760$	$11092755296$	$23720595868800$	$51385281922411832$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$158$	$2296$	$29074$	$372738$	$4830692$	$62765386$	$815806306$	$10604680408$	$137858134878$

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.ag $\times$ 2.13.ak_bx 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.ag : \(\Q(\sqrt{-1}) \).
• 2.13.ak_bx : 4.0.27200.2.

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.ae_c_bi	$2$	(not in LMFDB)
3.13.e_c_abi	$2$	(not in LMFDB)
3.13.q_es_vi	$2$	(not in LMFDB)