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

• Label: 3.11.ao_dq_aoy
• 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:	$( 1 - 5 x + 11 x^{2} )( 1 - 9 x + 38 x^{2} - 99 x^{3} + 121 x^{4} )$
	$1 - 14 x + 94 x^{2} - 388 x^{3} + 1034 x^{4} - 1694 x^{5} + 1331 x^{6}$
Frobenius angles:	$\pm0.0468428922585$, $\pm0.228229222880$, $\pm0.380176225592$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$364$	$1658384$	$2428264384$	$3141808488000$	$4160796310194524$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$114$	$1372$	$14658$	$160418$	$1769244$	$19484246$	$214351682$	$2357851972$	$25936927554$

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 $\F_{11^{6}}$.

Endomorphism algebra over $\F_{11}$

The isogeny class factors as 1.11.af $\times$ 2.11.aj_bm 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.aj_bm : \(\Q(\sqrt{-3}, \sqrt{17})\).

Endomorphism algebra over $\overline{\F}_{11}$

The base change of $A$ to $\F_{11^{6}}$ is 1.1771561.acna^2 $\times$ 1.1771561.bow. The endomorphism algebra for each factor is:

• 1.1771561.acna^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-51}) \)$)$
• 1.1771561.bow : \(\Q(\sqrt{-19}) \).

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{11^{2}}$
  The base change of $A$ to $\F_{11^{2}}$ is 1.121.ad $\times$ 2.121.af_ads. The endomorphism algebra for each factor is:

  • 1.121.ad : \(\Q(\sqrt{-19}) \).
  • 2.121.af_ads : \(\Q(\sqrt{-3}, \sqrt{17})\).
• Endomorphism algebra over $\F_{11^{3}}$
  The base change of $A$ to $\F_{11^{3}}$ is 1.1331.bo $\times$ 2.1331.a_acna. The endomorphism algebra for each factor is:

  • 1.1331.bo : \(\Q(\sqrt{-19}) \).
  • 2.1331.a_acna : \(\Q(\sqrt{-3}, \sqrt{17})\).

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.ae_e_ai	$2$	(not in LMFDB)
3.11.e_e_i	$2$	(not in LMFDB)
3.11.o_dq_oy	$2$	(not in LMFDB)
3.11.af_q_az	$3$	(not in LMFDB)