Abelian variety isogeny class 6.2.ag_p_at_g_bb_acj over $\F_{2}$

• Label: 6.2.ag_p_at_g_bb_acj
• Base field: $\F_{2}$
• Dimension: $6$
• $p$-rank: $6$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: no
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2}$
Dimension:	$6$
L-polynomial:	$1 - 6 x + 15 x^{2} - 19 x^{3} + 6 x^{4} + 27 x^{5} - 61 x^{6} + 54 x^{7} + 24 x^{8} - 152 x^{9} + 240 x^{10} - 192 x^{11} + 64 x^{12}$
Frobenius angles:	$\pm0.0163720992762$, $\pm0.0955680481608$, $\pm0.157549204067$, $\pm0.379771426289$, $\pm0.540012492605$, $\pm0.905260988165$
Angle rank:	$3$ (numerical)
Number field:	12.0.15342238784889.1
Galois group:	$C_2^2 \times A_4$
Jacobians:	$0$
Cyclic group of points:	yes

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$6$
Slopes:	$[0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1$	$577$	$195301$	$6045229$	$891243181$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-3$	$-1$	$6$	$-1$	$27$	$74$	$123$	$311$	$537$	$1049$

Jacobians and polarizations

This isogeny class is not principally polarizable, and therefore does not contain a Jacobian.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{9}}$.

Endomorphism algebra over $\F_{2}$

The endomorphism algebra of this simple isogeny class is 12.0.15342238784889.1.

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

The base change of $A$ to $\F_{2^{9}}$ is 3.512.m_aen_akdh^2 and its endomorphism algebra is $\mathrm{M}_{2}($6.0.1305639.1$)$

Remainder of endomorphism lattice by field

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

  The base change of $A$ to $\F_{2^{3}}$ is the simple isogeny class 6.8.ad_j_ak_p_ahk_sf and its endomorphism algebra is 12.0.15342238784889.1.

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
6.2.g_p_t_g_abb_acj	$2$	(not in LMFDB)
6.2.d_g_i_g_a_ah	$3$	(not in LMFDB)
6.2.d_g_i_p_bb_bv	$3$	(not in LMFDB)