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

• Label: 2.7.a_k
• 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:	$( 1 - 2 x + 7 x^{2} )( 1 + 2 x + 7 x^{2} )$
	$1 + 10 x^{2} + 49 x^{4}$
Frobenius angles:	$\pm0.376624142786$, $\pm0.623375857214$
Angle rank:	$1$ (numerical)
Jacobians:	$12$
Isomorphism classes:	40
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$60$	$3600$	$117180$	$5760000$	$282450300$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$70$	$344$	$2398$	$16808$	$116710$	$823544$	$5774398$	$40353608$	$282425350$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{7^{2}}$.

Endomorphism algebra over $\F_{7}$

The isogeny class factors as 1.7.ac $\times$ 1.7.c and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.7.ac : \(\Q(\sqrt{-6}) \).
• 1.7.c : \(\Q(\sqrt{-6}) \).

Endomorphism algebra over $\overline{\F}_{7}$

The base change of $A$ to $\F_{7^{2}}$ is 1.49.k^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.ae_s	$2$	2.49.u_hq
2.7.e_s	$2$	2.49.u_hq
2.7.a_ak	$4$	(not in LMFDB)