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

• Label: 2.7.ad_f
• 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:	$1 - 3 x + 5 x^{2} - 21 x^{3} + 49 x^{4}$
Frobenius angles:	$\pm0.130332681249$, $\pm0.613951744591$
Angle rank:	$2$ (numerical)
Number field:	4.0.2725.1
Galois group:	$D_{4}$
Jacobians:	$3$
Isomorphism classes:	3

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$31$	$2449$	$103261$	$5752701$	$288791536$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$5$	$51$	$299$	$2395$	$17180$	$117807$	$824213$	$5774179$	$40363463$	$282464286$

Jacobians and polarizations

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

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

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 4.0.2725.1.

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_f	$2$	2.49.b_ad