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

• Label: 2.17.ak_ch
• 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:	$( 1 - 5 x + 17 x^{2} )^{2}$
	$1 - 10 x + 59 x^{2} - 170 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.292637436158$, $\pm0.292637436158$
Angle rank:	$1$ (numerical)
Jacobians:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$169$	$89401$	$25441936$	$7059192361$	$2016777737689$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$308$	$5174$	$84516$	$1420408$	$24123422$	$410258584$	$6975597508$	$118588438358$	$2015999428628$

Jacobians and polarizations

This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{17}$

The isogeny class factors as 1.17.af^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-43}) \)$)$

Base change

This is a primitive isogeny class.

Twists

Below are some of the twists of this isogeny class.

Twist	Extension degree	Common base change
2.17.a_j	$2$	(not in LMFDB)
2.17.k_ch	$2$	(not in LMFDB)
2.17.f_i	$3$	(not in LMFDB)