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

• Label: 2.7.b_ae
• Base field: $\F_{7}$
• 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:	$\F_{7}$
Dimension:	$2$
L-polynomial:	$1 + x - 4 x^{2} + 7 x^{3} + 49 x^{4}$
Frobenius angles:	$\pm0.247407954773$, $\pm0.857771192050$
Angle rank:	$2$ (numerical)
Number field:	4.0.42632.1
Galois group:	$D_{4}$
Jacobians:	$3$
Isomorphism classes:	3

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$54$	$2052$	$130248$	$6057504$	$282999474$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$9$	$41$	$378$	$2521$	$16839$	$118298$	$820521$	$5764369$	$40335894$	$282480641$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which all are hyperelliptic):

• $y^2=3 x^6+6 x^5+3 x^4+4 x^2+4 x+1$
• $y^2=4 x^5+3 x^4+2 x+1$
• $y^2=x^6+5 x^5+x^4+3 x^3+3 x+3$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{7}$

The endomorphism algebra of this simple isogeny class is 4.0.42632.1.

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
2.7.ab_ae	$2$	2.49.aj_dw