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

• Label: 2.7.c_ad
• 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:	$1 + 2 x - 3 x^{2} + 14 x^{3} + 49 x^{4}$
Frobenius angles:	$\pm0.290042523881$, $\pm0.956709190548$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{2}, \sqrt{-3})\)
Galois group:	$C_2^2$
Jacobians:	$1$
Isomorphism classes:	3
Cyclic group of points:	no
Non-cyclic primes:	$3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$63$	$1953$	$142884$	$5767209$	$286584543$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$10$	$40$	$412$	$2404$	$17050$	$116710$	$822790$	$5760004$	$40362244$	$282500200$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is hyperelliptic):

• $y^2=x^6+x^3+4$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{7^{3}}$.

Endomorphism algebra over $\F_{7}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{2}, \sqrt{-3})\).

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

The base change of $A$ to $\F_{7^{3}}$ is 1.343.bi^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.ac_ad	$2$	2.49.ak_bz
2.7.ae_s	$3$	2.343.cq_csw
2.7.a_k	$6$	(not in LMFDB)