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

• Label: 6.2.a_c_a_c_a_a
• Base field: $\F_{2}$
• Dimension: $6$
• $p$-rank: $0$
• 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:	$1 + 2 x^{2} + 2 x^{4} + 8 x^{8} + 32 x^{10} + 64 x^{12}$
Frobenius angles:	$\pm0.112963726036$, $\pm0.310601149018$, $\pm0.423564875054$, $\pm0.576435124946$, $\pm0.689398850982$, $\pm0.887036273964$
Angle rank:	$2$ (numerical)
Number field:	12.0.319794774016000000.2
Galois group:	12T48
Jacobians:	$1$

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$109$	$11881$	$248629$	$19971961$	$1150235689$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$9$	$9$	$17$	$33$	$57$	$129$	$337$	$513$	$1169$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is not hyperelliptic), and hence is principally polarizable:

• $x^{9} + x^{7} y^{2} + x^{2} y^{7} + x^{6} y^{2} + x^{5} y^{3} + x^{6} y + x^{2} y^{5} + y^{7} + x^{5} y + x^{4} y^{2} + x^{2} y^{4} + y^{6} + x^{5} + x^{2} y^{3} + x y^{4} + y^{5} + x^{4} + x^{2} y^{2} + y^{4} + x^{3} + x^{2} y + y^{3} + y^{2} + x + y + 1=0$

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 12.0.319794774016000000.2.

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

The base change of $A$ to $\F_{2^{2}}$ is the simple isogeny class 6.4.e_i_i_u_ds_lc and its endomorphism algebra is the quaternion algebra over 6.0.17672000.2 with the following ramification data at primes above $2$, and unramified at all archimedean places:

$v$	($ 2 $,\( \frac{1}{16} \pi^{5} + \frac{5}{16} \pi^{3} + \frac{7}{4} \pi^{2} + \frac{5}{4} \pi + 1 \))	($ 2 $,\( \frac{61}{32} \pi^{5} + \frac{3}{16} \pi^{4} + \frac{45}{32} \pi^{3} + \frac{1}{16} \pi^{2} + \frac{5}{4} \pi + 1 \))	($ 2 $,\( \pi + 1 \))
$\operatorname{inv}_v$	$1/2$	$0$	$1/2$

where $\pi$ is a root of $x^{6} + 5x^{4} - 4x^{3} + 20x^{2} + 64$.

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_ac_a_c_a_a	$4$	(not in LMFDB)