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

• Label: 2.17.b_bc
• 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:	$( 1 - 2 x + 17 x^{2} )( 1 + 3 x + 17 x^{2} )$
	$1 + x + 28 x^{2} + 17 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.422020869623$, $\pm0.618522015261$
Angle rank:	$2$ (numerical)
Jacobians:	$21$
Cyclic group of points:	no
Non-cyclic primes:	$2$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$336$	$100800$	$23978304$	$6945120000$	$2016049835856$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$19$	$345$	$4882$	$83153$	$1419899$	$24132510$	$410358107$	$6975985633$	$118587178354$	$2015989341225$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 21 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{17}$

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

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_bo	$2$	(not in LMFDB)
2.17.ab_bc	$2$	(not in LMFDB)
2.17.f_bo	$2$	(not in LMFDB)