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

• Label: 2.7.ac_g
• 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:	$( 1 - 4 x + 7 x^{2} )( 1 + 2 x + 7 x^{2} )$
	$1 - 2 x + 6 x^{2} - 14 x^{3} + 49 x^{4}$
Frobenius angles:	$\pm0.227185525829$, $\pm0.623375857214$
Angle rank:	$2$ (numerical)
Jacobians:	$14$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$40$	$2880$	$112840$	$5990400$	$290600200$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$58$	$330$	$2494$	$17286$	$117466$	$822282$	$5765566$	$40336710$	$282428218$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{7}$

The isogeny class factors as 1.7.ae $\times$ 1.7.c 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.c : \(\Q(\sqrt{-6}) \).

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.ag_w	$2$	2.49.i_da
2.7.c_g	$2$	2.49.i_da
2.7.g_w	$2$	2.49.i_da
2.7.b_m	$3$	2.343.ao_g
2.7.h_y	$3$	2.343.ao_g