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

• Label: 2.7.ae_p
• 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:	$1 - 4 x + 15 x^{2} - 28 x^{3} + 49 x^{4}$
Frobenius angles:	$\pm0.250816204349$, $\pm0.483874642948$
Angle rank:	$2$ (numerical)
Number field:	4.0.56592.1
Galois group:	$D_{4}$
Jacobians:	$2$
Isomorphism classes:	2

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$33$	$3201$	$129492$	$5765001$	$284446833$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$64$	$376$	$2404$	$16924$	$118294$	$822868$	$5755588$	$40339240$	$282505504$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

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 4.0.56592.1.

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.e_p	$2$	2.49.o_dv