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

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

Invariants

Base field:	$\F_{7^{3}}$
Dimension:	$2$
L-polynomial:	$( 1 - 37 x + 343 x^{2} )( 1 - 32 x + 343 x^{2} )$
	$1 - 69 x + 1870 x^{2} - 23667 x^{3} + 117649 x^{4}$
Frobenius angles:	$\pm0.0148899108188$, $\pm0.168002921102$
Angle rank:	$2$ (numerical)

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$95784$	$13721632704$	$1627912484793504$	$191579706942413826816$	$22539338295385160791463064$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$275$	$116629$	$40341188$	$13841177065$	$4747561089665$	$1628413601042878$	$558545863939931471$	$191581231367828213809$	$65712362363046994120364$	$22539340290678694536133789$

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.abl $\times$ 1.343.abg and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.343.abl : \(\Q(\sqrt{-3}) \).
• 1.343.abg : \(\Q(\sqrt{-87}) \).

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
2.343.af_ate	$2$	(not in LMFDB)
2.343.f_ate	$2$	(not in LMFDB)
2.343.cr_cty	$2$	(not in LMFDB)
2.343.ap_fm	$3$	(not in LMFDB)
2.343.am_bu	$3$	(not in LMFDB)