Abelian variety isogeny class 4.3.e_i_q_bf over $\F_{3}$

• Label: 4.3.e_i_q_bf
• Base field: $\F_{3}$
• Dimension: $4$
• $p$-rank: $4$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{3}$
Dimension:	$4$
L-polynomial:	$1 + 4 x + 8 x^{2} + 16 x^{3} + 31 x^{4} + 48 x^{5} + 72 x^{6} + 108 x^{7} + 81 x^{8}$
Frobenius angles:	$\pm0.288152604671$, $\pm0.495448267664$, $\pm0.788152604671$, $\pm0.995448267664$
Angle rank:	$2$ (numerical)
Number field:	8.0.349241344.2
Galois group:	$D_4\times C_2$
Jacobians:	$0$
Cyclic group of points:	no
Non-cyclic primes:	$3$

This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$4$
Slopes:	$[0, 0, 0, 0, 1, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$369$	$6273$	$969732$	$39350529$	$3544437249$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$10$	$44$	$78$	$248$	$730$	$2192$	$6054$	$20276$	$59050$

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 $\F_{3^{4}}$.

Endomorphism algebra over $\F_{3}$

The endomorphism algebra of this simple isogeny class is 8.0.349241344.2.

Endomorphism algebra over $\overline{\F}_{3}$

The base change of $A$ to $\F_{3^{4}}$ is 2.81.ac_aev^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-9 -2 \sqrt{2}})\)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{3^{2}}$

  The base change of $A$ to $\F_{3^{2}}$ is the simple isogeny class 4.9.a_ac_a_aev and its endomorphism algebra is 8.0.349241344.2.

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
4.3.ae_i_aq_bf	$2$	(not in LMFDB)
4.3.ae_o_abg_cp	$8$	(not in LMFDB)
4.3.a_ag_a_t	$8$	(not in LMFDB)