Abelian variety isogeny class 2.2.c_c over $\F_{2}$

• Label: 2.2.c_c
• Base field: $\F_{2}$
• Dimension: $2$
• $p$-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:	$2$
L-polynomial:	$1 + 2 x + 2 x^{2} + 4 x^{3} + 4 x^{4}$
Frobenius angles:	$\pm0.416666666667$, $\pm0.916666666667$
Angle rank:	$0$ (numerical)
Number field:	\(\Q(\zeta_{12})\)
Galois group:	$C_2^2$
Jacobians:	$1$
Isomorphism classes:	2

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:	$[1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$13$	$13$	$169$	$169$	$793$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$5$	$5$	$17$	$9$	$25$	$65$	$145$	$289$	$449$	$1025$

Jacobians and polarizations

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

• $y^2+y=x^5+x^3$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{12}}$.

Endomorphism algebra over $\F_{2}$

The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{12})\).

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

The base change of $A$ to $\F_{2^{12}}$ is 1.4096.ey^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, 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 the simple isogeny class 2.4.a_ae and its endomorphism algebra is \(\Q(\zeta_{12})\).

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

  The base change of $A$ to $\F_{2^{3}}$ is 1.8.e^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

• Endomorphism algebra over $\F_{2^{4}}$

  The base change of $A$ to $\F_{2^{4}}$ is 1.16.ae^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

• Endomorphism algebra over $\F_{2^{6}}$

  The base change of $A$ to $\F_{2^{6}}$ is 1.64.a^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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
2.2.ac_c	$2$	2.4.a_ae
2.2.ae_i	$3$	2.8.i_bg
2.2.a_a	$6$	2.64.a_ey