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

• Label: 2.17.ak_by
• 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:	$( 1 - 8 x + 17 x^{2} )( 1 - 2 x + 17 x^{2} )$
	$1 - 10 x + 50 x^{2} - 170 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.0779791303774$, $\pm0.422020869623$
Angle rank:	$1$ (numerical)
Jacobians:	$10$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$160$	$83200$	$24088480$	$6922240000$	$2011667984800$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$8$	$290$	$4904$	$82878$	$1416808$	$24137570$	$410384584$	$6975884158$	$118587729608$	$2015993900450$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{17^{4}}$.

Endomorphism algebra over $\F_{17}$

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

Endomorphism algebra over $\overline{\F}_{17}$

The base change of $A$ to $\F_{17^{4}}$ is 1.83521.amk^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

Remainder of endomorphism lattice by field

• Endomorphism algebra over $\F_{17^{2}}$
  The base change of $A$ to $\F_{17^{2}}$ is 1.289.abe $\times$ 1.289.be. The endomorphism algebra for each factor is:

  • 1.289.abe : \(\Q(\sqrt{-1}) \).
  • 1.289.be : \(\Q(\sqrt{-1}) \).

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.ag_s	$2$	(not in LMFDB)
2.17.g_s	$2$	(not in LMFDB)
2.17.k_by	$2$	(not in LMFDB)