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

• Label: 3.13.aq_et_avp
• 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:	$( 1 - 5 x + 13 x^{2} )( 1 - 11 x + 55 x^{2} - 143 x^{3} + 169 x^{4} )$
	$1 - 16 x + 123 x^{2} - 561 x^{3} + 1599 x^{4} - 2704 x^{5} + 2197 x^{6}$
Frobenius angles:	$\pm0.129998747777$, $\pm0.256122854178$, $\pm0.292104599859$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$639$	$4601439$	$11225100852$	$23833986160959$	$51390434008460304$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$160$	$2323$	$29212$	$372773$	$4826557$	$62735440$	$815702932$	$10604639449$	$137859790175$

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.af $\times$ 2.13.al_cd 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.af : \(\Q(\sqrt{-3}) \).
• 2.13.al_cd : 4.0.6725.1.

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.ag_n_al	$2$	(not in LMFDB)
3.13.g_n_l	$2$	(not in LMFDB)
3.13.q_et_vp	$2$	(not in LMFDB)
3.13.an_dm_apg	$3$	(not in LMFDB)
3.13.ae_aj_dv	$3$	(not in LMFDB)