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

• Label: 2.17.al_cg
• 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:	$( 1 - 8 x + 17 x^{2} )( 1 - 3 x + 17 x^{2} )$
	$1 - 11 x + 58 x^{2} - 187 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.0779791303774$, $\pm0.381477984739$
Angle rank:	$2$ (numerical)
Jacobians:	$3$
Isomorphism classes:	21

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$150$	$81900$	$24242400$	$6945120000$	$2011609803750$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$7$	$285$	$4936$	$83153$	$1416767$	$24130530$	$410365151$	$6975985633$	$118588427752$	$2015993714925$

Jacobians and polarizations

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

• $y^2=3x^6+16x^5+4x^4+2x^3+6x^2+5$
• $y^2=11x^6+11x^5+9x^4+3x^3+3x^2+4x+7$
• $y^2=7x^6+11x^5+3x^4+6x^3+15x^2+3$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{17}$

The isogeny class factors as 1.17.ai $\times$ 1.17.ad 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.ai : \(\Q(\sqrt{-1}) \).
• 1.17.ad : \(\Q(\sqrt{-59}) \).

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.af_k	$2$	(not in LMFDB)
2.17.f_k	$2$	(not in LMFDB)
2.17.l_cg	$2$	(not in LMFDB)