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

• Label: 3.7.a_f_q
• 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:	$1 + 5 x^{2} + 16 x^{3} + 35 x^{4} + 343 x^{6}$
Frobenius angles:	$\pm0.281765751916$, $\pm0.434672644429$, $\pm0.815654334581$
Angle rank:	$3$ (numerical)
Number field:	6.0.5169344.1
Galois group:	$S_4\times C_2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$400$	$147200$	$46510000$	$14366720000$	$4635764242000$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$60$	$392$	$2492$	$16408$	$118140$	$822424$	$5762172$	$40348904$	$282458300$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 54 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:

• $y^2=x^7+x^5+6 x+6$
• $y^2=x^7+x^5+5 x^3+4 x+5$
• $y^2=x^7+3 x^5+x^2+4 x$
• $y^2=x^7+3 x^5+4 x^2+2 x+2$
• $y^2=x^7+3 x^5+x^3+4 x^2+4 x+1$
• $y^2=x^7+3 x^5+2 x^3+x^2+1$
• $y^2=x^7+3 x^5+2 x^3+3 x^2+3 x+2$
• $y^2=x^7+3 x^5+2 x^3+4 x^2+3 x$
• $y^2=x^7+3 x^5+4 x^3+3 x^2+2 x+3$
• $y^2=x^7+3 x^5+4 x^3+6 x^2+1$
• $y^2=x^7+x^6+x^3+5 x^2+x+5$
• $y^2=x^7+x^6+6 x^3+5 x^2+4 x$
• $y^2=x^7+x^6+2 x^4+5 x^3+6 x^2+6 x+1$
• $y^2=x^7+x^6+2 x^4+6 x^3+3 x^2+4 x+2$
• $y^2=x^7+x^6+3 x^4+4 x^2+3 x$
• $y^2=x^7+x^6+3 x^4+x^3+4 x+2$
• $y^2=x^7+x^6+3 x^4+2 x^3+2 x^2+2$
• $y^2=x^7+x^6+3 x^4+2 x^3+6 x^2+2 x+5$
• $y^2=x^7+x^6+4 x^4+4 x^3+4 x^2+2 x+1$
• $y^2=x^7+x^6+5 x^4+2 x^3+5 x^2+2 x$
• and

• $y^2=x^7+x^6+6 x^4+6 x^3+5 x^2+6 x+1$
• $y^2=x^7+3 x^6+3 x^4+6 x^3+3 x^2+2 x+2$
• $y^2=x^7+3 x^6+4 x^4+3 x^3+6 x^2+3 x+4$
• $y^2=x^8+x^4+2 x^3+2 x^2+6 x+3$
• $y^2=x^8+3 x^4+2 x^3+5 x^2+2 x+5$
• $y^2=x^8+x^5+3 x^3+6 x^2+4 x+4$
• $y^2=x^8+x^5+x^4+2 x^3+x^2+4 x+2$
• $y^2=x^8+x^5+3 x^4+4 x^3+x^2+5 x+2$
• $y^2=x^8+x^5+3 x^4+6 x^3+x^2+6 x+4$
• $y^2=x^8+2 x^5+3 x^3+x^2+6 x+5$
• $y^2=x^8+2 x^5+3 x^3+2 x^2+3 x+2$
• $y^2=x^8+2 x^5+x^4+2 x^2+5 x+2$
• $y^2=x^8+2 x^5+x^4+3 x^3+5 x^2+4 x+6$
• $y^2=x^8+2 x^5+x^4+5 x^3+x^2+4 x+6$
• $y^2=x^8+2 x^5+3 x^4+5 x^2+5 x+3$
• $y^2=x^8+2 x^5+3 x^4+x^3+x^2+2 x+3$
• $y^2=x^8+2 x^5+3 x^4+x^3+3 x^2+x+1$
• $y^2=x^8+3 x^5+x^4+4 x^2+5 x+1$
• $y^2=x^8+3 x^5+x^4+2 x^3+x+5$
• $y^2=x^8+3 x^5+x^4+4 x^3+x+3$
• $y^2=x^8+3 x^5+3 x^4+4 x^2+2 x+2$
• $y^2=x^8+x^6+x^3+x^2+2$
• $y^2=x^8+x^6+x^4+3 x^3+4 x^2+x+5$
• $y^2=x^8+x^6+6 x^4+2 x^3+4 x^2+3 x+5$
• $y^2=x^8+x^6+x^5+4 x^3+2 x^2+2 x+5$
• $y^2=x^8+x^6+x^5+2 x^4+5 x^3+4 x^2+3$
• $y^2=x^8+x^6+x^5+2 x^4+6 x^3+x+6$
• $y^2=x^8+x^6+x^5+4 x^4+x^3+4 x^2+2 x+6$
• $y^2=x^8+x^6+x^5+5 x^4+5 x^2+6 x+3$
• $y^2=x^8+x^6+2 x^5+x^4+x^3+3 x^2+2 x+5$
• $y^2=x^8+x^6+2 x^5+6 x^4+x^3+5 x^2+x+6$
• $y^2=x^8+x^6+3 x^5+6 x^2+3 x+6$
• $y^2=x^8+x^6+3 x^5+2 x^4+x^3+6 x^2+3 x+2$
• $y^2=x^8+3 x^6+4 x^4+2 x^3+5 x^2+x+2$

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 6.0.5169344.1.

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.a_f_aq	$2$	(not in LMFDB)