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

• Label: 3.11.ao_dn_aoe
• 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:	$( 1 - 6 x + 11 x^{2} )( 1 - 8 x + 32 x^{2} - 88 x^{3} + 121 x^{4} )$
	$1 - 14 x + 91 x^{2} - 368 x^{3} + 1001 x^{4} - 1694 x^{5} + 1331 x^{6}$
Frobenius angles:	$\pm0.0750991438595$, $\pm0.140218899004$, $\pm0.424900856141$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$348$	$1566000$	$2313872532$	$3088152000000$	$4162394387139708$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$108$	$1306$	$14404$	$160478$	$1773900$	$19505498$	$214404284$	$2357982046$	$25937522028$

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 $\F_{11^{4}}$.

Endomorphism algebra over $\F_{11}$

The isogeny class factors as 1.11.ag $\times$ 2.11.ai_bg 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 : \(\Q(\sqrt{-2}) \).
• 2.11.ai_bg : \(\Q(i, \sqrt{6})\).

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

The base change of $A$ to $\F_{11^{4}}$ is 1.14641.afm^2 $\times$ 1.14641.bu. The endomorphism algebra for each factor is:

• 1.14641.afm^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-6}) \)$)$
• 1.14641.bu : \(\Q(\sqrt{-2}) \).

Remainder of endomorphism lattice by field

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

  • 1.121.ao : \(\Q(\sqrt{-2}) \).
  • 2.121.a_afm : \(\Q(i, \sqrt{6})\).

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.ac_af_q	$2$	(not in LMFDB)
3.11.c_af_aq	$2$	(not in LMFDB)
3.11.o_dn_oe	$2$	(not in LMFDB)