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

• Label: 2.17.a_abe
• 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:	$( 1 - 8 x + 17 x^{2} )( 1 + 8 x + 17 x^{2} )$
	$1 - 30 x^{2} + 289 x^{4}$
Frobenius angles:	$\pm0.0779791303774$, $\pm0.922020869623$
Angle rank:	$1$ (numerical)
Jacobians:	$5$

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$260$	$67600$	$24136580$	$6922240000$	$2015996087300$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$18$	$230$	$4914$	$82878$	$1419858$	$24135590$	$410338674$	$6975884158$	$118587876498$	$2015998274150$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{17^{2}}$.

Endomorphism algebra over $\F_{17}$

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

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

The base change of $A$ to $\F_{17^{2}}$ is 1.289.abe^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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.aq_du	$2$	(not in LMFDB)
2.17.q_du	$2$	(not in LMFDB)
2.17.ak_by	$4$	(not in LMFDB)