Abelian variety isogeny class 2.8.h_bc over $\F_{2^{3}}$

• Label: 2.8.h_bc
• Base field: $\F_{2^{3}}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: no

Invariants

Base field:	$\F_{2^{3}}$
Dimension:	$2$
L-polynomial:	$( 1 + 3 x + 8 x^{2} )( 1 + 4 x + 8 x^{2} )$
	$1 + 7 x + 28 x^{2} + 56 x^{3} + 64 x^{4}$
Frobenius angles:	$\pm0.677932000632$, $\pm0.750000000000$
Angle rank:	$1$ (numerical)
Jacobians:	$0$

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

Newton polygon

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$156$	$4680$	$225108$	$17643600$	$1069417596$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$72$	$436$	$4304$	$32636$	$261144$	$2101220$	$16770976$	$134212108$	$1073792232$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

Decomposition and endomorphism algebra

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

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

The isogeny class factors as 1.8.d $\times$ 1.8.e and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.8.d : \(\Q(\sqrt{-23}) \).
• 1.8.e : \(\Q(\sqrt{-1}) \).

Endomorphism algebra over $\overline{\F}_{2^{3}}$

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

• 1.4096.db : \(\Q(\sqrt{-23}) \).
• 1.4096.ey : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{2^{6}}$
  The base change of $A$ to $\F_{2^{6}}$ is 1.64.a $\times$ 1.64.h. The endomorphism algebra for each factor is:

  • 1.64.a : \(\Q(\sqrt{-1}) \).
  • 1.64.h : \(\Q(\sqrt{-23}) \).

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
2.8.ah_bc	$2$	2.64.h_ey
2.8.ab_e	$2$	2.64.h_ey
2.8.b_e	$2$	2.64.h_ey