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

• Label: 3.13.ap_ec_arz
• 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:	$( 1 - 7 x + 13 x^{2} )( 1 - 8 x + 37 x^{2} - 104 x^{3} + 169 x^{4} )$
	$1 - 15 x + 106 x^{2} - 467 x^{3} + 1378 x^{4} - 2535 x^{5} + 2197 x^{6}$
Frobenius angles:	$\pm0.0772104791556$, $\pm0.167453355204$, $\pm0.421338968608$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$665$	$4454835$	$10580245760$	$23132465785275$	$51114869396254125$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$157$	$2192$	$28357$	$370779$	$4831084$	$62779751$	$815801509$	$10604449616$	$137858083157$

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.ai_bl 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.ai_bl : 4.0.16025.1.

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.ab_ag_bz	$2$	(not in LMFDB)
3.13.b_ag_abz	$2$	(not in LMFDB)
3.13.p_ec_rz	$2$	(not in LMFDB)
3.13.ag_bi_afe	$3$	(not in LMFDB)
3.13.ad_k_ax	$3$	(not in LMFDB)