Abelian variety isogeny class 3.7.a_f_s over $\F_{7}$

• Label: 3.7.a_f_s
• Base field: $\F_{7}$
• Dimension: $3$
• $p$-rank: $3$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{7}$
Dimension:	$3$
L-polynomial:	$1 + 5 x^{2} + 18 x^{3} + 35 x^{4} + 343 x^{6}$
Frobenius angles:	$\pm0.291094381517$, $\pm0.424344988283$, $\pm0.820776537321$
Angle rank:	$3$ (numerical)
Number field:	6.0.216207903168.1
Galois group:	$S_4\times C_2$
Isomorphism classes:	40
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:	$3$
Slopes:	$[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$402$	$147132$	$47290878$	$14370088176$	$4621979732442$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$60$	$398$	$2492$	$16358$	$117936$	$822284$	$5764892$	$40351988$	$282456600$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 5 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:

• $y^2=x^8+2 x^5+x^4+6 x^3+6 x^2+5 x+6$
• $y^2=x^8+3 x^5+3 x^4+6 x^3+3 x^2+2 x+6$
• $y^2=x^8+x^6+3 x^3+3 x^2+x+3$
• $y^2=x^8+x^6+x^5+5 x^3+6 x^2+5 x+6$
• $y^2=x^8+x^6+2 x^5+3 x^4+6 x^3+4 x^2+x+1$

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 6.0.216207903168.1.

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
3.7.a_f_as	$2$	(not in LMFDB)