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

• Label: 2.17.aj_bp
• Base field: $\F_{17}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• 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:	$1 - 9 x + 41 x^{2} - 153 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.0511302365929$, $\pm0.466745019538$
Angle rank:	$2$ (numerical)
Number field:	4.0.328653.3
Galois group:	$D_{4}$
Jacobians:	$4$
Isomorphism classes:	4
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:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$169$	$83317$	$23743993$	$6893398629$	$2011960631824$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$9$	$291$	$4833$	$82531$	$1417014$	$24139923$	$410348241$	$6975598339$	$118587234249$	$2015995427166$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):

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

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 4.0.328653.3.

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.j_bp	$2$	(not in LMFDB)