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

• Label: 2.7.ae_s
• Base field: $\F_{7}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{7}$
Dimension:	$2$
L-polynomial:	$( 1 - 2 x + 7 x^{2} )^{2}$
	$1 - 4 x + 18 x^{2} - 28 x^{3} + 49 x^{4}$
Frobenius angles:	$\pm0.376624142786$, $\pm0.376624142786$
Angle rank:	$1$ (numerical)
Jacobians:	$3$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3$

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$36$	$3600$	$142884$	$5760000$	$274432356$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$4$	$70$	$412$	$2398$	$16324$	$116710$	$825052$	$5774398$	$40362244$	$282425350$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{7}$

The isogeny class factors as 1.7.ac^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-6}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.7.a_k	$2$	2.49.u_hq
2.7.e_s	$2$	2.49.u_hq
2.7.c_ad	$3$	2.343.cq_csw