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

• Label: 2.17.e_bd
• 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:	$( 1 - x + 17 x^{2} )( 1 + 5 x + 17 x^{2} )$
	$1 + 4 x + 29 x^{2} + 68 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.461304015105$, $\pm0.707362563842$
Angle rank:	$2$ (numerical)
Jacobians:	$11$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$391$	$96577$	$23747776$	$6974501209$	$2013674226751$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$22$	$332$	$4834$	$83508$	$1418222$	$24137822$	$410409182$	$6975583396$	$118586983618$	$2015997651932$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 11 curves (of which all are hyperelliptic):

• $y^2=13 x^6+x^5+13 x^4+8 x^3+13 x^2+x+13$
• $y^2=2 x^6+16 x^5+x^4+14 x^3+x^2+16 x+2$
• $y^2=2 x^6+5 x^5+10 x^4+14 x^3+14 x^2+7 x+1$
• $y^2=8 x^6+14 x^5+13 x^4+13 x^3+6 x^2+12 x+13$
• $y^2=15 x^6+15 x^5+8 x^4+12 x^3+8 x^2+15 x+15$
• $y^2=7 x^6+2 x+9$
• $y^2=14 x^6+14 x^5+6 x^4+8 x^3+2 x^2+4 x+1$
• $y^2=16 x^6+3 x^5+2 x^4+8 x^3+6 x^2+8 x+11$
• $y^2=6 x^6+3 x^5+4 x^4+9 x^3+7 x^2+5 x+16$
• $y^2=5 x^6+13 x^5+10 x^4+5 x^3+5 x^2+10 x+12$
• $y^2=13 x^6+16 x^5+12 x^4+5 x^3+2 x^2+13 x+15$

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{17}$

The isogeny class factors as 1.17.ab $\times$ 1.17.f 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.ab : \(\Q(\sqrt{-67}) \).
• 1.17.f : \(\Q(\sqrt{-43}) \).

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.ag_bn	$2$	(not in LMFDB)
2.17.ae_bd	$2$	(not in LMFDB)
2.17.g_bn	$2$	(not in LMFDB)