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

• Label: 2.17.ak_cd
• 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:	$( 1 - 7 x + 17 x^{2} )( 1 - 3 x + 17 x^{2} )$
	$1 - 10 x + 55 x^{2} - 170 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.177280642489$, $\pm0.381477984739$
Angle rank:	$2$ (numerical)
Jacobians:	$18$
Isomorphism classes:	30

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$165$	$86625$	$24837120$	$7001465625$	$2015920649325$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$300$	$5054$	$83828$	$1419808$	$24141150$	$410388784$	$6975964708$	$118587816158$	$2015989381500$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{17}$

The isogeny class factors as 1.17.ah $\times$ 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.ah : \(\Q(\sqrt{-19}) \).
• 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.ae_n	$2$	(not in LMFDB)
2.17.e_n	$2$	(not in LMFDB)
2.17.k_cd	$2$	(not in LMFDB)