Abelian variety isogeny class 2.7.ah_y over $\F_{7}$

• Label: 2.7.ah_y
• Base field: $\F_{7}$
• 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:	$\F_{7}$
Dimension:	$2$
L-polynomial:	$( 1 - 5 x + 7 x^{2} )( 1 - 2 x + 7 x^{2} )$
	$1 - 7 x + 24 x^{2} - 49 x^{3} + 49 x^{4}$
Frobenius angles:	$\pm0.106147807505$, $\pm0.376624142786$
Angle rank:	$2$ (numerical)
Jacobians:	$1$
Isomorphism classes:	6

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$18$	$2340$	$122472$	$5709600$	$278855478$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$1$	$49$	$358$	$2377$	$16591$	$117466$	$825553$	$5773873$	$40370506$	$282483289$

Jacobians and polarizations

This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:

• $y^2=3x^6+3x^5+x^4+2x^3+5x+5$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{7}$

The isogeny class factors as 1.7.af $\times$ 1.7.ac and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.7.af : \(\Q(\sqrt{-3}) \).
• 1.7.ac : \(\Q(\sqrt{-6}) \).

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
2.7.ad_e	$2$	2.49.ab_am
2.7.d_e	$2$	2.49.ab_am
2.7.h_y	$2$	2.49.ab_am
2.7.ab_m	$3$	2.343.o_g
2.7.c_g	$3$	2.343.o_g