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

• Label: 3.13.ap_dx_aqq
• 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:	$( 1 - 7 x + 13 x^{2} )( 1 - 8 x + 32 x^{2} - 104 x^{3} + 169 x^{4} )$
	$1 - 15 x + 101 x^{2} - 432 x^{3} + 1313 x^{4} - 2535 x^{5} + 2197 x^{6}$
Frobenius angles:	$\pm0.0370621216586$, $\pm0.0772104791556$, $\pm0.462937878341$
Angle rank:	$2$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$630$	$4154220$	$10012857120$	$22657863639600$	$50895254892840750$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$147$	$2072$	$27767$	$369179$	$4826304$	$62751191$	$815683199$	$10604336456$	$137859052107$

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 $\F_{13^{4}}$.

Endomorphism algebra over $\F_{13}$

The isogeny class factors as 1.13.ah $\times$ 2.13.ai_bg 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_bg : \(\Q(i, \sqrt{10})\).

Endomorphism algebra over $\overline{\F}_{13}$

The base change of $A$ to $\F_{13^{4}}$ is 1.28561.alq^2 $\times$ 1.28561.ahj. The endomorphism algebra for each factor is:

• 1.28561.alq^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-10}) \)$)$
• 1.28561.ahj : \(\Q(\sqrt{-3}) \).

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{13^{2}}$
  The base change of $A$ to $\F_{13^{2}}$ is 1.169.ax $\times$ 2.169.a_alq. The endomorphism algebra for each factor is:

  • 1.169.ax : \(\Q(\sqrt{-3}) \).
  • 2.169.a_alq : \(\Q(i, \sqrt{10})\).

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_al_q	$2$	(not in LMFDB)
3.13.b_al_aq	$2$	(not in LMFDB)
3.13.p_dx_qq	$2$	(not in LMFDB)
3.13.ag_bd_afo	$3$	(not in LMFDB)
3.13.ad_f_abw	$3$	(not in LMFDB)