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

• Label: 2.7.f_q
• 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:	$1 + 5 x + 16 x^{2} + 35 x^{3} + 49 x^{4}$
Frobenius angles:	$\pm0.526405026161$, $\pm0.830821217411$
Angle rank:	$2$ (numerical)
Number field:	4.0.57800.1
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$106$	$2756$	$114904$	$5666336$	$280383886$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$13$	$57$	$334$	$2361$	$16683$	$118938$	$821029$	$5763153$	$40363138$	$282478897$

Jacobians and polarizations

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

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

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 4.0.57800.1.

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.af_q	$2$	2.49.h_e