Abelian variety isogeny class 2.343.o_kd over $\F_{7^{3}}$

• Label: 2.343.o_kd
• Base field: $\F_{7^{3}}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{7^{3}}$
Dimension:	$2$
L-polynomial:	$1 + 14 x + 263 x^{2} + 4802 x^{3} + 117649 x^{4}$
Frobenius angles:	$\pm0.369859557165$, $\pm0.782510844020$
Angle rank:	$2$ (numerical)
Number field:	\(\Q(\sqrt{-851 +28 \sqrt{118}})\)
Galois group:	$D_{4}$
Cyclic group of points:	yes

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$122729$	$13880281713$	$1628659927012400$	$191584430437509477369$	$22539301964574977961646529$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$358$	$117980$	$40359712$	$13841518324$	$4747553437138$	$1628413582009070$	$558545864601073966$	$191581231389073760164$	$65712362364305114519776$	$22539340290678613328681900$

Jacobians and polarizations

This isogeny class contains a Jacobian and hence is principally polarizable.

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{7^{3}}$.

Endomorphism algebra over $\F_{7^{3}}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-851 +28 \sqrt{118}})\).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.343.ao_kd	$2$	(not in LMFDB)