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

• Label: 3.11.ap_eb_aqw
• 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:	$( 1 - 3 x + 11 x^{2} )( 1 - 6 x + 11 x^{2} )^{2}$
	$1 - 15 x + 105 x^{2} - 438 x^{3} + 1155 x^{4} - 1815 x^{5} + 1331 x^{6}$
Frobenius angles:	$\pm0.140218899004$, $\pm0.140218899004$, $\pm0.350615407277$
Angle rank:	$2$ (numerical)
Cyclic group of points:	no
Non-cyclic primes:	$2, 3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$324$	$1574640$	$2424140784$	$3174575016960$	$4187001208875804$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-3$	$107$	$1368$	$14807$	$161427$	$1773716$	$19503537$	$214437167$	$2358165528$	$25937613227$

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.ad 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.ad : \(\Q(\sqrt{-35}) \).

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.aj_bh_adm	$2$	(not in LMFDB)
3.11.ad_ad_bq	$2$	(not in LMFDB)
3.11.d_ad_abq	$2$	(not in LMFDB)
3.11.j_bh_dm	$2$	(not in LMFDB)
3.11.p_eb_qw	$2$	(not in LMFDB)
3.11.d_s_cf	$3$	(not in LMFDB)