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

• Label: 2.17.am_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:	$( 1 - 7 x + 17 x^{2} )( 1 - 5 x + 17 x^{2} )$
	$1 - 12 x + 69 x^{2} - 204 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.177280642489$, $\pm0.292637436158$
Angle rank:	$2$ (numerical)
Jacobians:	$3$
Isomorphism classes:	4

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$143$	$82225$	$24856832$	$7047093625$	$2019558358103$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$6$	$284$	$5058$	$84372$	$1422366$	$24140126$	$410328078$	$6975719908$	$118587953826$	$2015994517964$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{17}$

The isogeny class factors as 1.17.ah $\times$ 1.17.af 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.ah : \(\Q(\sqrt{-19}) \).
• 1.17.af : \(\Q(\sqrt{-43}) \).

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