Abelian variety isogeny class 3.11.ar_ez_avi over $\F_{11}$

• Label: 3.11.ar_ez_avi
• Base field: $\F_{11}$
• 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:	$\F_{11}$
Dimension:	$3$
L-polynomial:	$( 1 - 5 x + 11 x^{2} )( 1 - 6 x + 11 x^{2} )^{2}$
	$1 - 17 x + 129 x^{2} - 554 x^{3} + 1419 x^{4} - 2057 x^{5} + 1331 x^{6}$
Frobenius angles:	$\pm0.140218899004$, $\pm0.140218899004$, $\pm0.228229222880$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$252$	$1388016$	$2368889712$	$3209092992000$	$4220866870857252$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-5$	$91$	$1336$	$14967$	$162725$	$1777300$	$19502135$	$214388207$	$2357984056$	$25937415931$

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 $\F_{11}$.

Endomorphism algebra over $\F_{11}$

The isogeny class factors as 1.11.ag^2 $\times$ 1.11.af and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.11.ag^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$
• 1.11.af : \(\Q(\sqrt{-19}) \).

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
3.11.ah_j_ba	$2$	(not in LMFDB)
3.11.af_ad_cs	$2$	(not in LMFDB)
3.11.f_ad_acs	$2$	(not in LMFDB)
3.11.h_j_aba	$2$	(not in LMFDB)
3.11.r_ez_vi	$2$	(not in LMFDB)
3.11.b_g_h	$3$	(not in LMFDB)