Abelian variety isogeny class 3.7.e_p_bm over $\F_{7}$

• Label: 3.7.e_p_bm
• Base field: $\F_{7}$
• Dimension: $3$
• $p$-rank: $3$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{7}$
Dimension:	$3$
L-polynomial:	$1 + 4 x + 15 x^{2} + 38 x^{3} + 105 x^{4} + 196 x^{5} + 343 x^{6}$
Frobenius angles:	$\pm0.360544620689$, $\pm0.610915698974$, $\pm0.816693384916$
Angle rank:	$3$ (numerical)
Number field:	6.0.166682304.1
Galois group:	$S_4\times C_2$
Cyclic group of points:	no
Non-cyclic primes:	$3$

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:	$3$
Slopes:	$[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$702$	$158652$	$40075074$	$14238699696$	$4689132370182$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$12$	$64$	$342$	$2468$	$16602$	$117364$	$821364$	$5774300$	$40368564$	$282403564$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 10 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:

• $y^2=x^8+3 x^4+3 x^3+4 x^2+3 x+1$
• $y^2=x^8+x^5+x^4+2 x^3+6 x^2+6 x+5$
• $y^2=x^8+x^5+x^4+5 x^3+4 x^2+4 x+6$
• $y^2=x^8+2 x^5+x^3+4 x^2+6 x+2$
• $y^2=x^8+2 x^5+x^3+5 x^2+1$
• $y^2=x^8+2 x^5+x^3+5 x^2+6 x+5$
• $y^2=x^8+x^6+2 x^4+x^3+2 x^2+3 x+6$
• $y^2=x^8+x^6+x^5+2 x^3+4 x+6$
• $y^2=x^8+x^6+x^5+5 x^4+6 x^3+3 x^2+6$
• $y^2=x^8+x^6+x^5+5 x^4+6 x^3+6 x^2+3 x+2$

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 6.0.166682304.1.

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
3.7.ae_p_abm	$2$	(not in LMFDB)