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

• Label: 2.17.ae_bf
• 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:	$1 - 4 x + 31 x^{2} - 68 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.309500678080$, $\pm0.524952027357$
Angle rank:	$2$ (numerical)
Number field:	4.0.2459408.1
Galois group:	$D_{4}$
Jacobians:	$10$
Isomorphism classes:	10
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$249$	$97857$	$24653988$	$6965167689$	$2016785858649$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$14$	$336$	$5018$	$83396$	$1420414$	$24137478$	$410282446$	$6975609604$	$118588861274$	$2015998625616$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):

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

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 4.0.2459408.1.

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.e_bf	$2$	(not in LMFDB)