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

• Label: 3.13.ar_fc_axf
• 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:	$( 1 - 7 x + 13 x^{2} )( 1 - 10 x + 49 x^{2} - 130 x^{3} + 169 x^{4} )$
	$1 - 17 x + 132 x^{2} - 603 x^{3} + 1716 x^{4} - 2873 x^{5} + 2197 x^{6}$
Frobenius angles:	$\pm0.0772104791556$, $\pm0.151058869957$, $\pm0.334339837461$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$553$	$4169067$	$10652248768$	$23367257826171$	$51161662525287013$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-3$	$145$	$2208$	$28645$	$371117$	$4826116$	$62758497$	$815826469$	$10604911584$	$137859378185$

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.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.ah : \(\Q(\sqrt{-3}) \).
• 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.ad_ai_df	$2$	(not in LMFDB)
3.13.d_ai_adf	$2$	(not in LMFDB)
3.13.r_fc_xf	$2$	(not in LMFDB)
3.13.ai_bq_agg	$3$	(not in LMFDB)
3.13.af_m_ap	$3$	(not in LMFDB)