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

• Label: 2.17.am_cq
• 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:	$1 - 12 x + 68 x^{2} - 204 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.144218155731$, $\pm0.312293972145$
Angle rank:	$2$ (numerical)
Number field:	4.0.39168.3
Galois group:	$D_{4}$
Jacobians:	$4$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$142$	$81508$	$24674062$	$7021751184$	$2017512450862$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$282$	$5022$	$84070$	$1420926$	$24137466$	$410345382$	$6975904510$	$118588855302$	$2015997043962$

Jacobians and polarizations

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

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

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 4.0.39168.3.

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.m_cq	$2$	(not in LMFDB)