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

• Label: 2.7.a_ac
• 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:	$( 1 - 4 x + 7 x^{2} )( 1 + 4 x + 7 x^{2} )$
	$1 - 2 x^{2} + 49 x^{4}$
Frobenius angles:	$\pm0.227185525829$, $\pm0.772814474171$
Angle rank:	$1$ (numerical)
Jacobians:	$7$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$48$	$2304$	$117936$	$6230016$	$282453168$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$46$	$344$	$2590$	$16808$	$118222$	$823544$	$5756734$	$40353608$	$282431086$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 7 curves (of which all are hyperelliptic):

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

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.ae $\times$ 1.7.e 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.ae : \(\Q(\sqrt{-3}) \).
• 1.7.e : \(\Q(\sqrt{-3}) \).

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

The base change of $A$ to $\F_{7^{2}}$ is 1.49.ac^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$

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.ai_be	$2$	2.49.ae_dy
2.7.i_be	$2$	2.49.ae_dy
2.7.aj_bi	$3$	2.343.a_la
2.7.ag_t	$3$	2.343.a_la
2.7.ad_k	$3$	2.343.a_la
2.7.a_al	$3$	2.343.a_la
2.7.a_n	$3$	2.343.a_la
2.7.d_k	$3$	2.343.a_la
2.7.g_t	$3$	2.343.a_la
2.7.j_bi	$3$	2.343.a_la