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

• Label: 3.2.ad_d_ac
• Base field: $\F_{2}$
• Dimension: $3$
• $p$-rank: $2$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{2}$
Dimension:	$3$
L-polynomial:	$( 1 - 2 x + 2 x^{2} )( 1 - x - x^{2} - 2 x^{3} + 4 x^{4} )$
	$1 - 3 x + 3 x^{2} - 2 x^{3} + 6 x^{4} - 12 x^{5} + 8 x^{6}$
Frobenius angles:	$\pm0.0516399385854$, $\pm0.250000000000$, $\pm0.718306605252$
Angle rank:	$1$ (numerical)
Jacobians:	$1$
Isomorphism classes:	3

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1$	$35$	$208$	$6475$	$30791$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$2$	$3$	$26$	$30$	$47$	$126$	$194$	$471$	$1082$

Jacobians and polarizations

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

• $x^4+x^3y+x^3z+x^2yz+y^4+y^3z+z^4=0$

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.ac $\times$ 2.2.ab_ab 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.ac : \(\Q(\sqrt{-1}) \).
• 2.2.ab_ab : \(\Q(\sqrt{-3}, \sqrt{-7})\).

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

The base change of $A$ to $\F_{2^{12}}$ is 1.4096.bv^2 $\times$ 1.4096.ey. The endomorphism algebra for each factor is:

• 1.4096.bv^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
• 1.4096.ey : 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 $\times$ 2.4.ad_f. The endomorphism algebra for each factor is:

  • 1.4.a : \(\Q(\sqrt{-1}) \).
  • 2.4.ad_f : \(\Q(\sqrt{-3}, \sqrt{-7})\).
• Endomorphism algebra over $\F_{2^{3}}$
  The base change of $A$ to $\F_{2^{3}}$ is 1.8.af^2 $\times$ 1.8.e. The endomorphism algebra for each factor is:

  • 1.8.af^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
  • 1.8.e : \(\Q(\sqrt{-1}) \).
• Endomorphism algebra over $\F_{2^{4}}$
  The base change of $A$ to $\F_{2^{4}}$ is 1.16.i $\times$ 2.16.b_ap. The endomorphism algebra for each factor is:

  • 1.16.i : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
  • 2.16.b_ap : \(\Q(\sqrt{-3}, \sqrt{-7})\).
• Endomorphism algebra over $\F_{2^{6}}$
  The base change of $A$ to $\F_{2^{6}}$ is 1.64.aj^2 $\times$ 1.64.a. The endomorphism algebra for each factor is:

  • 1.64.aj^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-7}) \)$)$
  • 1.64.a : \(\Q(\sqrt{-1}) \).

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
3.2.ab_ab_g	$2$	3.4.ad_j_ay
3.2.b_ab_ag	$2$	3.4.ad_j_ay
3.2.d_d_c	$2$	3.4.ad_j_ay
3.2.a_d_ac	$3$	3.8.ag_j_e

Additional information

This is the isogeny class of the Jacobian of a function field of class number 1. It also appears as a sporadic example in the classification of abelian varieties with one rational point.