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

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

Invariants

Base field:	$\F_{7^{3}}$
Dimension:	$2$
L-polynomial:	$( 1 - 36 x + 343 x^{2} )^{2}$
	$1 - 72 x + 1982 x^{2} - 24696 x^{3} + 117649 x^{4}$
Frobenius angles:	$\pm0.0756263964363$, $\pm0.0756263964363$
Angle rank:	$1$ (numerical)

This isogeny class is not simple, not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$94864$	$13698361600$	$1627638013248016$	$191577444395420160000$	$22539324833061912486832144$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$272$	$116430$	$40334384$	$13841013598$	$4747558254032$	$1628413574543790$	$558545864358862064$	$191581231398501988798$	$65712362364085437544592$	$22539340290705947852427150$

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 isogeny class factors as 1.343.abk^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-19}) \)$)$

Base change

This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{7^{3}}$.

Subfield	Primitive Model
$\F_{7}$	2.7.ad_c
$\F_{7}$	2.7.g_x

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.343.a_axm	$2$	(not in LMFDB)
2.343.cu_cyg	$2$	(not in LMFDB)
2.343.bk_bkr	$3$	(not in LMFDB)