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

• Label: 2.17.am_cs
• 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:	$( 1 - 6 x + 17 x^{2} )^{2}$
	$1 - 12 x + 70 x^{2} - 204 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.240632536990$, $\pm0.240632536990$
Angle rank:	$1$ (numerical)
Jacobians:	$6$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$144$	$82944$	$25040016$	$7072137216$	$2021435619984$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$286$	$5094$	$84670$	$1423686$	$24141022$	$410294310$	$6975432574$	$118586681478$	$2015992253086$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{17}$

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

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_ac	$2$	(not in LMFDB)
2.17.m_cs	$2$	(not in LMFDB)
2.17.g_t	$3$	(not in LMFDB)