Abelian variety isogeny class 3.17.aj_cf_ajt over $\F_{17}$

• Label: 3.17.aj_cf_ajt
• Base field: $\F_{17}$
• 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:	$\F_{17}$
Dimension:	$3$
L-polynomial:	$1 - 9 x + 57 x^{2} - 253 x^{3} + 969 x^{4} - 2601 x^{5} + 4913 x^{6}$
Frobenius angles:	$\pm0.169376685505$, $\pm0.345923157333$, $\pm0.575735532212$
Angle rank:	$3$ (numerical)
Number field:	6.0.392918607.1
Galois group:	$A_4\times C_2$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$3077$	$27086831$	$119829829517$	$583868804192119$	$2867972070538621117$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$9$	$323$	$4965$	$83699$	$1422609$	$24139235$	$410322264$	$6976009859$	$118589048961$	$2015990585483$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 18 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:

• $y^2=3 x^7+4 x^5+10 x^4+5 x^2+6$
• $y^2=3 x^7+3 x^5+15 x^4+14 x^3+11 x^2+x+7$
• $y^2=3 x^7+9 x^5+7 x^4+16 x^3+13 x+10$
• $y^2=x^7+10 x^5+13 x^4+5 x^3+16 x^2+10 x+12$
• $y^2=x^7+12 x^5+4 x^4+12 x^3+x^2+3 x+12$
• $y^2=x^7+5 x^5+2 x^3+9 x+12$
• $y^2=x^7+8 x^5+2 x^4+4 x^3+10 x^2+3 x+11$
• $y^2=x^7+4 x^5+2 x^4+14 x^3+x^2+7 x+12$
• $y^2=x^7+12 x^5+4 x^4+6 x^3+14 x^2+16 x+7$
• $y^2=3 x^7+13 x^5+11 x^4+10 x^3+15 x^2+3$
• $y^2=x^7+9 x^5+12 x^4+9 x^3+15 x^2+2$
• $y^2=x^7+11 x^5+4 x^4+12 x^3+15 x^2+11 x+3$
• $y^2=3 x^7+5 x^5+5 x^4+4 x^3+11 x^2+10 x+1$
• $y^2=x^7+13 x^5+10 x^4+5 x^3+8 x^2+x+6$
• $y^2=x^7+12 x^5+14 x^4+7 x^3+5 x^2+14 x+1$
• $y^2=x^7+14 x^5+3 x^4+x^3+3 x^2+x+6$
• $y^2=3 x^7+2 x^5+6 x^4+6 x^3+8 x^2+3 x+11$
• $y^2=x^7+9 x^5+14 x^4+x^3+3 x^2+8 x+12$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{17}$.

Endomorphism algebra over $\F_{17}$

The endomorphism algebra of this simple isogeny class is 6.0.392918607.1.

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
3.17.j_cf_jt	$2$	(not in LMFDB)