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

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

Invariants

Base field:	$\F_{2}$
Dimension:	$6$
L-polynomial:	$1 + x^{2} - 2 x^{6} + 16 x^{10} + 64 x^{12}$
Frobenius angles:	$\pm0.0884648938753$, $\pm0.282996084564$, $\pm0.425920832374$, $\pm0.574079167626$, $\pm0.717003915436$, $\pm0.911535106125$
Angle rank:	$3$ (numerical)
Number field:	12.0.5722438253412096409600.1
Galois group:	12T139
Jacobians:	$3$

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

Newton polygon

$p$-rank:	$2$
Slopes:	$[0, 0, 1/4, 1/4, 1/4, 1/4, 3/4, 3/4, 3/4, 3/4, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$80$	$6400$	$251120$	$16000000$	$1149544400$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$7$	$9$	$15$	$33$	$55$	$129$	$271$	$513$	$1167$

Jacobians and polarizations

This isogeny class contains the Jacobians of 3 curves (of which 0 are hyperelliptic), and hence is principally polarizable:

• $x^{4} y^{3} + x^{4} y^{2} + x^{3} y^{2} + x^{2} y^{3} + x^{4} + x^{3} + x^{2} y + y^{3} + y^{2} + y=0$
• $ x^{9} + x^{8} y + x^{6} y^{3} + x^{3} y^{6} + x^{2} y^{7} + x^{6} y^{2} + x^{5} y^{3} + x^{2} y^{6} + x^{7} + x^{5} y^{2} + x^{2} y^{5} + x y^{6} + y^{7} + x^{6} + x^{5} y + x^{3} y^{3} + x^{2} y^{4} + x y^{5} + x^{5} + x^{4} y + x^{3} y^{2} + x y^{4} + y^{5} + x^{4} + x y^{3} + x y^{2} + x^{2} + x y + x + y=0$
• $ x^{10} + x^{8} y^{2} + x^{6} y^{4} + x^{8} y + x^{6} y^{3} + x^{5} y + x^{4} y^{2} + x^{3} y^{3} + x^{2} y^{4} + x^{5} + x^{4} y + x^{3} y^{2} + x^{2} y^{3} + x^{4} + y^{4} + x^{2} + y^{2}=0$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{2}$

The endomorphism algebra of this simple isogeny class is 12.0.5722438253412096409600.1.

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

The base change of $A$ to $\F_{2^{2}}$ is the simple isogeny class 6.4.c_b_ae_ae_bg_gi and its endomorphism algebra is the quaternion algebra over 6.0.2363962480.1 with the following ramification data at primes above $2$, and unramified at all archimedean places:

$v$	($ 2 $,\( \pi + 1 \))	($ 2 $,\( \frac{1}{2} \pi^{3} + \frac{3}{2} \pi^{2} + \pi + 1 \))	($ 2 $,\( \frac{31}{16} \pi^{5} + \frac{1}{16} \pi^{4} + \frac{1}{2} \pi^{3} + \frac{11}{8} \pi^{2} + 1 \))	($ 2 $,\( \frac{1}{16} \pi^{5} + \frac{3}{16} \pi^{4} + \frac{7}{4} \pi^{3} + \frac{1}{8} \pi^{2} + \frac{1}{2} \pi \))
$\operatorname{inv}_v$	$0$	$1/2$	$1/2$	$0$

where $\pi$ is a root of $x^{6} - x^{5} + 2x^{3} - 16x + 64$.

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
6.2.a_ab_a_a_a_c	$4$	(not in LMFDB)