Abelian variety isogeny class 5.2.ab_a_ac_b_d over $\F_{2}$

• Label: 5.2.ab_a_ac_b_d
• Base field: $\F_{2}$
• Dimension: $5$
• $p$-rank: $5$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2}$
Dimension:	$5$
L-polynomial:	$1 - x - 2 x^{3} + x^{4} + 3 x^{5} + 2 x^{6} - 8 x^{7} - 16 x^{9} + 32 x^{10}$
Frobenius angles:	$\pm0.0443041605502$, $\pm0.217315895900$, $\pm0.516999344923$, $\pm0.644724456345$, $\pm0.871889092795$
Angle rank:	$5$ (numerical)
Number field:	10.0.111140444342016.1
Galois group:	$C_2 \wr S_5$
Jacobians:	$2$
Isomorphism classes:	75
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$5$
Slopes:	$[0, 0, 0, 0, 0, 1, 1, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$12$	$720$	$13716$	$748800$	$45045012$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$2$	$4$	$2$	$12$	$42$	$76$	$86$	$252$	$398$	$1044$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which 1 is hyperelliptic):

• $y^2 + (x^4 + x^3 + x^2 + x + 1) x y = x^{12} + x^6 + x^3 + x^2 + 1$
• $x z + y^2 + y z = x^2 + x z + y t + y u + z t + z u + t u = x z + y^2 + y z + y t + z^2 + z t + z u + t^2 + u^2 = 0$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{2}$

The endomorphism algebra of this simple isogeny class is 10.0.111140444342016.1.

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
5.2.b_a_c_b_ad	$2$	(not in LMFDB)