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

• Label: 2.2.c_e
• Base field: $\F_{2}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2}$
Dimension:	$2$
L-polynomial:	$( 1 + 2 x^{2} )( 1 + 2 x + 2 x^{2} )$
	$1 + 2 x + 4 x^{2} + 4 x^{3} + 4 x^{4}$
Frobenius angles:	$\pm0.5$, $\pm0.750000000000$
Angle rank:	$0$ (numerical)
Jacobians:	$1$
Isomorphism classes:	3

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$15$	$45$	$45$	$225$	$825$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$5$	$9$	$5$	$17$	$25$	$81$	$145$	$193$	$545$	$1089$

Jacobians and polarizations

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

• $y^2+y=x^5+x^4$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{2^{8}}$.

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 1.2.a $\times$ 1.2.c and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.2.a : \(\Q(\sqrt{-2}) \).
• 1.2.c : \(\Q(\sqrt{-1}) \).

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

The base change of $A$ to $\F_{2^{8}}$ is 1.256.abg^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 1.4.a $\times$ 1.4.e. The endomorphism algebra for each factor is:

  • 1.4.a : \(\Q(\sqrt{-1}) \).
  • 1.4.e : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• Endomorphism algebra over $\F_{2^{4}}$
  The base change of $A$ to $\F_{2^{4}}$ is 1.16.ai $\times$ 1.16.i. The endomorphism algebra for each factor is:

  • 1.16.ai : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 1.16.i : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

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_e	$2$	2.4.e_i
2.2.ae_i	$8$	2.256.acm_chc
2.2.a_ae	$8$	2.256.acm_chc