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

• Label: 3.11.ap_ee_arn
• 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:	$( 1 - 5 x + 11 x^{2} )^{3}$
	$1 - 15 x + 108 x^{2} - 455 x^{3} + 1188 x^{4} - 1815 x^{5} + 1331 x^{6}$
Frobenius angles:	$\pm0.228229222880$, $\pm0.228229222880$, $\pm0.228229222880$
Angle rank:	$1$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$343$	$1685159$	$2582630848$	$3291326171875$	$4233994921204433$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-3$	$113$	$1452$	$15341$	$163227$	$1774748$	$19479177$	$214283861$	$2357660532$	$25936814033$

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^3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-19}) \)$)$

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.af_i_p	$2$	(not in LMFDB)
3.11.f_i_ap	$2$	(not in LMFDB)
3.11.p_ee_rn	$2$	(not in LMFDB)
3.11.a_a_bo	$3$	(not in LMFDB)