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

• Label: 4.2.a_c_a_e
• Base field: $\F_{2}$
• Dimension: $4$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2}$
Dimension:	$4$
L-polynomial:	$1 + 2 x^{2} + 4 x^{4} + 8 x^{6} + 16 x^{8}$
Frobenius angles:	$\pm0.200000000000$, $\pm0.400000000000$, $\pm0.600000000000$, $\pm0.800000000000$
Angle rank:	$0$ (numerical)
Number field:	8.0.64000000.2
Galois group:	$C_4\times C_2$
Jacobians:	$1$
Isomorphism classes:	18

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

Newton polygon

This isogeny class is supersingular.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$31$	$961$	$4681$	$116281$	$923521$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$9$	$9$	$25$	$33$	$81$	$129$	$289$	$513$	$769$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

• $y^2 + y=x^9 + x^5 + x$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{10}}$.

Endomorphism algebra over $\F_{2}$

The endomorphism algebra of this simple isogeny class is 8.0.64000000.2.

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

The base change of $A$ to $\F_{2^{10}}$ is 1.1024.acm^4 and its endomorphism algebra is $\mathrm{M}_{4}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

Remainder of endomorphism lattice by field

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

  The base change of $A$ to $\F_{2^{2}}$ is 2.4.c_e^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\zeta_{5})\)$)$

• Endomorphism algebra over $\F_{2^{5}}$

  The base change of $A$ to $\F_{2^{5}}$ is 2.32.a_acm^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q(\sqrt{2}) \) ramified at both real infinite places.

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.a_ac_a_e	$4$	(not in LMFDB)
4.2.a_ai_a_y	$5$	(not in LMFDB)
4.2.ac_c_a_ae	$8$	(not in LMFDB)