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

• Label: 2.7.ac_p
• 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:	$( 1 - x + 7 x^{2} )^{2}$
	$1 - 2 x + 15 x^{2} - 14 x^{3} + 49 x^{4}$
Frobenius angles:	$\pm0.439481140838$, $\pm0.439481140838$
Angle rank:	$1$ (numerical)
Jacobians:	$2$
Cyclic group of points:	no
Non-cyclic primes:	$7$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$49$	$3969$	$132496$	$5433561$	$275460409$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$76$	$384$	$2260$	$16386$	$118222$	$827070$	$5764324$	$40328448$	$282453436$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{7}$

The isogeny class factors as 1.7.ab^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.a_n	$2$	2.49.ba_kh
2.7.c_p	$2$	2.49.ba_kh
2.7.ai_be	$3$	2.343.bo_bpu
2.7.af_s	$3$	2.343.bo_bpu
2.7.b_ag	$3$	2.343.bo_bpu
2.7.e_j	$3$	2.343.bo_bpu
2.7.k_bn	$3$	2.343.bo_bpu