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

• Label: 2.17.ak_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:	$( 1 - 6 x + 17 x^{2} )( 1 - 4 x + 17 x^{2} )$
	$1 - 10 x + 58 x^{2} - 170 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.240632536990$, $\pm0.338793663197$
Angle rank:	$2$ (numerical)
Jacobians:	$8$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$168$	$88704$	$25290216$	$7045226496$	$2016775051368$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$306$	$5144$	$84350$	$1420408$	$24129522$	$410300584$	$6975697534$	$118587959528$	$2015994055986$

Jacobians and polarizations

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

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{17}$

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

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_k	$2$	(not in LMFDB)
2.17.c_k	$2$	(not in LMFDB)
2.17.k_cg	$2$	(not in LMFDB)