Abelian variety isogeny class 2.17.ac_r over $\F_{17}$

• Label: 2.17.ac_r
• Base field: $\F_{17}$
• Dimension: $2$
• $p$-rank: $1$
• Ordinary: no
• Supersingular: no
• Simple: yes
• Geometrically simple: yes
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{17}$
Dimension:	$2$
L-polynomial:	$1 - 2 x + 17 x^{2} - 34 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.280685333850$, $\pm0.628641673003$
Angle rank:	$2$ (numerical)
Number field:	4.0.149056.1
Galois group:	$D_{4}$
Jacobians:	$18$
Cyclic group of points:	yes

This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

$p$-rank:	$1$
Slopes:	$[0, 1/2, 1/2, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$271$	$92953$	$24091900$	$7022878009$	$2020056222271$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$320$	$4906$	$84084$	$1422716$	$24124790$	$410285948$	$6975804004$	$118587605482$	$2015994467600$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):

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

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 4.0.149056.1.

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
2.17.c_r	$2$	(not in LMFDB)