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

• Label: 2.7.ab_ag
• 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:	$( 1 - 5 x + 7 x^{2} )( 1 + 4 x + 7 x^{2} )$
	$1 - x - 6 x^{2} - 7 x^{3} + 49 x^{4}$
Frobenius angles:	$\pm0.106147807505$, $\pm0.772814474171$
Angle rank:	$1$ (numerical)
Jacobians:	$2$
Isomorphism classes:	24

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$36$	$1872$	$104976$	$5937984$	$278956476$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$7$	$37$	$304$	$2473$	$16597$	$118222$	$825307$	$5765041$	$40378768$	$282486157$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{7^{3}}$.

Endomorphism algebra over $\F_{7}$

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

Endomorphism algebra over $\overline{\F}_{7}$

The base change of $A$ to $\F_{7^{3}}$ is 1.343.au^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.aj_bi	$2$	2.49.an_eq
2.7.b_ag	$2$	2.49.an_eq
2.7.j_bi	$2$	2.49.an_eq
2.7.ak_bn	$3$	2.343.abo_bpu
2.7.ae_j	$3$	2.343.abo_bpu
2.7.c_p	$3$	2.343.abo_bpu
2.7.f_s	$3$	2.343.abo_bpu
2.7.i_be	$3$	2.343.abo_bpu