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

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

Invariants

Base field:	$\F_{17}$
Dimension:	$2$
L-polynomial:	$( 1 - 6 x + 17 x^{2} )( 1 - 3 x + 17 x^{2} )$
	$1 - 9 x + 52 x^{2} - 153 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.240632536990$, $\pm0.381477984739$
Angle rank:	$2$ (numerical)
Jacobians:	$9$
Isomorphism classes:	45

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$180$	$90720$	$25220160$	$7019913600$	$2015468442900$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$9$	$313$	$5130$	$84049$	$1419489$	$24133246$	$410337153$	$6975759841$	$118587422250$	$2015990704393$

Jacobians and polarizations

This isogeny class contains the Jacobians of 9 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{17}$

The isogeny class factors as 1.17.ag $\times$ 1.17.ad and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.17.ag : \(\Q(\sqrt{-2}) \).
• 1.17.ad : \(\Q(\sqrt{-59}) \).

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.ad_q	$2$	(not in LMFDB)
2.17.d_q	$2$	(not in LMFDB)
2.17.j_ca	$2$	(not in LMFDB)