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

• Label: 2.17.m_cr
• 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:	$( 1 + 5 x + 17 x^{2} )( 1 + 7 x + 17 x^{2} )$
	$1 + 12 x + 69 x^{2} + 204 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.707362563842$, $\pm0.822719357511$
Angle rank:	$2$ (numerical)
Jacobians:	$3$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$575$	$82225$	$23441600$	$7047093625$	$2012436350375$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$30$	$284$	$4770$	$84372$	$1417350$	$24140126$	$410349270$	$6975719908$	$118587799170$	$2015994517964$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{17}$

The isogeny class factors as 1.17.f $\times$ 1.17.h and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.17.f : \(\Q(\sqrt{-43}) \).
• 1.17.h : \(\Q(\sqrt{-19}) \).

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.am_cr	$2$	(not in LMFDB)
2.17.ac_ab	$2$	(not in LMFDB)
2.17.c_ab	$2$	(not in LMFDB)