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

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

Invariants

Base field:	$\F_{7}$
Dimension:	$2$
L-polynomial:	$1 - 3 x + 12 x^{2} - 21 x^{3} + 49 x^{4}$
Frobenius angles:	$\pm0.264975166102$, $\pm0.533843891264$
Angle rank:	$2$ (numerical)
Number field:	4.0.122536.2
Galois group:	$D_{4}$
Jacobians:	$4$
Isomorphism classes:	4

This isogeny class is simple and 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$38$	$3268$	$124184$	$5777824$	$287014418$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$5$	$65$	$362$	$2409$	$17075$	$118010$	$820685$	$5757169$	$40359494$	$282506825$

Jacobians and polarizations

This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{7}$

The endomorphism algebra of this simple isogeny class is 4.0.122536.2.

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.d_m	$2$	2.49.p_em