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

• Label: 2.17.aj_bz
• 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:	$1 - 9 x + 51 x^{2} - 153 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.223072927732$, $\pm0.393933150683$
Angle rank:	$2$ (numerical)
Number field:	4.0.290173.1
Galois group:	$D_{4}$
Jacobians:	$4$
Isomorphism classes:	4

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$179$	$90037$	$25084523$	$7010010709$	$2015786994224$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$9$	$311$	$5103$	$83931$	$1419714$	$24138263$	$410360211$	$6975775699$	$118587090069$	$2015988986366$

Jacobians and polarizations

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

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

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 4.0.290173.1.

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.j_bz	$2$	(not in LMFDB)