Abelian variety isogeny class 3.2.a_e_a over $\F_{2}$

• Label: 3.2.a_e_a
• Base field: $\F_{2}$
• Dimension: $3$
• $p$-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:	$3$
L-polynomial:	$( 1 + 2 x^{2} )( 1 + 2 x^{2} + 4 x^{4} )$
	$1 + 4 x^{2} + 8 x^{4} + 8 x^{6}$
Frobenius angles:	$\pm0.333333333333$, $\pm0.5$, $\pm0.666666666667$
Angle rank:	$0$ (numerical)
Jacobians:	$1$

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$21$	$441$	$441$	$3969$	$34881$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$3$	$13$	$9$	$17$	$33$	$49$	$129$	$257$	$513$	$1153$

Jacobians and polarizations

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

• $x^4+x^3 y+x z^3+y^4=0$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{2}$

The isogeny class factors as 1.2.a $\times$ 2.2.a_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 : \(\Q(\sqrt{-2}) \).
• 2.2.a_c : \(\Q(\sqrt{2}, \sqrt{-3})\).

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

The base change of $A$ to $\F_{2^{12}}$ is 1.4096.aey^3 and its endomorphism algebra is $\mathrm{M}_{3}(B)$, 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.c^2 $\times$ 1.4.e. The endomorphism algebra for each factor is:

  • 1.4.c^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
  • 1.4.e : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
• Endomorphism algebra over $\F_{2^{3}}$
  The base change of $A$ to $\F_{2^{3}}$ is 1.8.a $\times$ 2.8.a_aq. The endomorphism algebra for each factor is:

  • 1.8.a : \(\Q(\sqrt{-2}) \).
  • 2.8.a_aq : the quaternion algebra over \(\Q(\sqrt{2}) \) ramified at both real infinite places.
• Endomorphism algebra over $\F_{2^{4}}$
  The base change of $A$ to $\F_{2^{4}}$ is 1.16.ai $\times$ 1.16.e^2. The endomorphism algebra for each factor is:

  • 1.16.ai : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 1.16.e^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
• Endomorphism algebra over $\F_{2^{6}}$
  The base change of $A$ to $\F_{2^{6}}$ is 1.64.aq^2 $\times$ 1.64.q. The endomorphism algebra for each factor is:

  • 1.64.aq^2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 1.64.q : 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
3.2.a_ac_a	$3$	3.8.a_ai_a
3.2.a_a_a	$4$	3.16.a_a_aey
3.2.ae_i_am	$8$	(not in LMFDB)