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

• Label: 2.7.a_ae
• 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:	$1 - 4 x^{2} + 49 x^{4}$
Frobenius angles:	$\pm0.203884584447$, $\pm0.796115415553$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{2}, \sqrt{-5})\)
Galois group:	$C_2^2$
Jacobians:	$3$
Isomorphism classes:	6
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$46$	$2116$	$118174$	$6170256$	$282441886$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$42$	$344$	$2566$	$16808$	$118698$	$823544$	$5760958$	$40353608$	$282408522$

Jacobians and polarizations

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

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

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 \(\Q(\sqrt{2}, \sqrt{-5})\).

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

The base change of $A$ to $\F_{7^{2}}$ is 1.49.ae^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-5}) \)$)$

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
2.7.a_e	$4$	(not in LMFDB)
2.7.ag_s	$8$	(not in LMFDB)
2.7.g_s	$8$	(not in LMFDB)