Abelian variety isogeny class 4.2.ae_f_c_al over $\F_{2}$

• Label: 4.2.ae_f_c_al
• Base field: $\F_{2}$
• 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:	$\F_{2}$
Dimension:	$4$
L-polynomial:	$1 - 4 x + 5 x^{2} + 2 x^{3} - 11 x^{4} + 4 x^{5} + 20 x^{6} - 32 x^{7} + 16 x^{8}$
Frobenius angles:	$\pm0.0247483856139$, $\pm0.177336015878$, $\pm0.344002682545$, $\pm0.858081718947$
Angle rank:	$2$ (numerical)
Number field:	8.0.22581504.2
Galois group:	$D_4\times C_2$
Jacobians:	$0$
Isomorphism classes:	1

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1$	$61$	$5044$	$76921$	$660661$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$-1$	$11$	$19$	$19$	$65$	$97$	$291$	$479$	$959$

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 $\F_{2^{12}}$.

Endomorphism algebra over $\F_{2}$

The endomorphism algebra of this simple isogeny class is 8.0.22581504.2.

Endomorphism algebra over $\overline{\F}_{2}$

The base change of $A$ to $\F_{2^{12}}$ is 2.4096.ahm_zkj^2 and its endomorphism algebra is $\mathrm{M}_{2}($4.0.4752.1$)$

Remainder of endomorphism lattice by field

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

  The base change of $A$ to $\F_{2^{2}}$ is the simple isogeny class 4.4.ag_t_abq_dd and its endomorphism algebra is 8.0.22581504.2.

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

  The base change of $A$ to $\F_{2^{3}}$ is the simple isogeny class 4.8.c_c_ak_aep and its endomorphism algebra is 8.0.22581504.2.

• Endomorphism algebra over $\F_{2^{4}}$

  The base change of $A$ to $\F_{2^{4}}$ is the simple isogeny class 4.16.c_t_adq_adr and its endomorphism algebra is 8.0.22581504.2.

• Endomorphism algebra over $\F_{2^{6}}$

  The base change of $A$ to $\F_{2^{6}}$ is the simple isogeny class 4.64.a_ahm_a_zkj and its endomorphism algebra is 8.0.22581504.2.

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
4.2.e_f_ac_al	$2$	4.4.ag_t_abq_dd
4.2.c_c_c_b	$3$	(not in LMFDB)
4.2.c_f_i_n	$3$	(not in LMFDB)