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

• Label: 3.13.as_fq_abac
• 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:	$( 1 - 7 x + 13 x^{2} )( 1 - 6 x + 13 x^{2} )( 1 - 5 x + 13 x^{2} )$
	$1 - 18 x + 146 x^{2} - 678 x^{3} + 1898 x^{4} - 3042 x^{5} + 2197 x^{6}$
Frobenius angles:	$\pm0.0772104791556$, $\pm0.187167041811$, $\pm0.256122854178$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$504$	$4021920$	$10695089664$	$23612933635200$	$51398916541701624$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-4$	$138$	$2216$	$28946$	$372836$	$4829868$	$62746820$	$815695394$	$10604386568$	$137858511018$

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 $\F_{13^{6}}$.

Endomorphism algebra over $\F_{13}$

The isogeny class factors as 1.13.ah $\times$ 1.13.ag $\times$ 1.13.af 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 : \(\Q(\sqrt{-3}) \).
• 1.13.ag : \(\Q(\sqrt{-1}) \).
• 1.13.af : \(\Q(\sqrt{-3}) \).

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

The base change of $A$ to $\F_{13^{6}}$ is 1.4826809.atm^2 $\times$ 1.4826809.gao. The endomorphism algebra for each factor is:

• 1.4826809.atm^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
• 1.4826809.gao : \(\Q(\sqrt{-1}) \).

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 $\times$ 1.169.ak $\times$ 1.169.b. The endomorphism algebra for each factor is:

  • 1.169.ax : \(\Q(\sqrt{-3}) \).
  • 1.169.ak : \(\Q(\sqrt{-1}) \).
  • 1.169.b : \(\Q(\sqrt{-3}) \).
• Endomorphism algebra over $\F_{13^{3}}$
  The base change of $A$ to $\F_{13^{3}}$ is 1.2197.acs $\times$ 1.2197.s $\times$ 1.2197.cs. The endomorphism algebra for each factor is:

  • 1.2197.acs : \(\Q(\sqrt{-3}) \).
  • 1.2197.s : \(\Q(\sqrt{-1}) \).
  • 1.2197.cs : \(\Q(\sqrt{-3}) \).

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.ai_q_c	$2$	(not in LMFDB)
3.13.ag_c_cc	$2$	(not in LMFDB)
3.13.ae_ai_ec	$2$	(not in LMFDB)
3.13.e_ai_aec	$2$	(not in LMFDB)
3.13.g_c_acc	$2$	(not in LMFDB)
3.13.i_q_ac	$2$	(not in LMFDB)
3.13.s_fq_bac	$2$	(not in LMFDB)
3.13.ap_ed_asg	$3$	(not in LMFDB)
3.13.aj_bv_ags	$3$	(not in LMFDB)
3.13.ag_ak_fi	$3$	(not in LMFDB)
3.13.ag_o_ag	$3$	(not in LMFDB)
3.13.ag_bj_afc	$3$	(not in LMFDB)
3.13.ad_l_as	$3$	(not in LMFDB)
3.13.d_ab_ag	$3$	(not in LMFDB)
3.13.g_c_acc	$3$	(not in LMFDB)