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

• Label: 2.17.k_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:	$( 1 + 4 x + 17 x^{2} )( 1 + 6 x + 17 x^{2} )$
	$1 + 10 x + 58 x^{2} + 170 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.661206336803$, $\pm0.759367463010$
Angle rank:	$2$ (numerical)
Jacobians:	$8$
Isomorphism classes:	24
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$528$	$88704$	$23029776$	$7045226496$	$2015213207568$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$28$	$306$	$4684$	$84350$	$1419308$	$24129522$	$410376764$	$6975697534$	$118587793468$	$2015994055986$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{17}$

The isogeny class factors as 1.17.e $\times$ 1.17.g 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.e : \(\Q(\sqrt{-13}) \).
• 1.17.g : \(\Q(\sqrt{-2}) \).

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.ak_cg	$2$	(not in LMFDB)
2.17.ac_k	$2$	(not in LMFDB)
2.17.c_k	$2$	(not in LMFDB)