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

• Label: 3.11.aq_en_atd
• 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:	$( 1 - 5 x + 11 x^{2} )( 1 - 11 x + 51 x^{2} - 121 x^{3} + 121 x^{4} )$
	$1 - 16 x + 117 x^{2} - 497 x^{3} + 1287 x^{4} - 1936 x^{5} + 1331 x^{6}$
Frobenius angles:	$\pm0.0215640055172$, $\pm0.228229222880$, $\pm0.270299311731$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$287$	$1488095$	$2408204372$	$3188429549375$	$4185769049802032$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-4$	$100$	$1361$	$14876$	$161381$	$1769185$	$19468326$	$214283236$	$2357751971$	$25937161875$

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.af $\times$ 2.11.al_bz 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.af : \(\Q(\sqrt{-19}) \).
• 2.11.al_bz : \(\Q(\zeta_{5})\).

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.ag_h_n	$2$	(not in LMFDB)
3.11.g_h_an	$2$	(not in LMFDB)
3.11.q_en_td	$2$	(not in LMFDB)