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

• Label: 2.7.ac_ab
• 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:	$( 1 - 5 x + 7 x^{2} )( 1 + 3 x + 7 x^{2} )$
	$1 - 2 x - x^{2} - 14 x^{3} + 49 x^{4}$
Frobenius angles:	$\pm0.106147807505$, $\pm0.691875465479$
Angle rank:	$2$ (numerical)
Jacobians:	$4$
Isomorphism classes:	12

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$33$	$2145$	$99792$	$5888025$	$283484553$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$44$	$288$	$2452$	$16866$	$117326$	$826398$	$5768548$	$40356576$	$282540764$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{7}$

The isogeny class factors as 1.7.af $\times$ 1.7.d 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.af : \(\Q(\sqrt{-3}) \).
• 1.7.d : \(\Q(\sqrt{-19}) \).

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_bd	$2$	2.49.ag_br
2.7.c_ab	$2$	2.49.ag_br
2.7.i_bd	$2$	2.49.ag_br
2.7.e_r	$3$	2.343.ace_ccc
2.7.h_ba	$3$	2.343.ace_ccc