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

• Label: 2.7.c_n
• 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:	$1 + 2 x + 13 x^{2} + 14 x^{3} + 49 x^{4}$
Frobenius angles:	$\pm0.475057501380$, $\pm0.650805391100$
Angle rank:	$2$ (numerical)
Number field:	4.0.39488.1
Galois group:	$D_{4}$
Jacobians:	$3$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$79$	$3713$	$108388$	$5617769$	$283869199$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$10$	$72$	$316$	$2340$	$16890$	$117606$	$824806$	$5764740$	$40333924$	$282499912$

Jacobians and polarizations

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

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

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 4.0.39488.1.

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.ac_n	$2$	2.49.w_id