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

• Label: 2.17.m_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:	$\pm0.759367463010$, $\pm0.759367463010$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$576$	$82944$	$23270976$	$7072137216$	$2010565187136$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$30$	$286$	$4734$	$84670$	$1416030$	$24141022$	$410383038$	$6975432574$	$118589071518$	$2015992253086$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{17}$

The isogeny class factors as 1.17.g^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.am_cs	$2$	(not in LMFDB)
2.17.a_ac	$2$	(not in LMFDB)
2.17.ag_t	$3$	(not in LMFDB)