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

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

Invariants

Base field:	$\F_{7}$
Dimension:	$2$
L-polynomial:	$1 - 10 x^{2} + 49 x^{4}$
Frobenius angles:	$\pm0.123375857214$, $\pm0.876624142786$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{6})\)
Galois group:	$C_2^2$
Jacobians:	$3$
Isomorphism classes:	12

This isogeny class is simple but not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$40$	$1600$	$118120$	$5760000$	$282500200$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$30$	$344$	$2398$	$16808$	$118590$	$823544$	$5774398$	$40353608$	$282525150$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{7}$

The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{6})\).

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

The base change of $A$ to $\F_{7^{2}}$ is 1.49.ak^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	$4$	(not in LMFDB)
2.7.a_k	$4$	(not in LMFDB)
2.7.e_s	$4$	(not in LMFDB)