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

• Label: 2.17.a_j
• 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:	$( 1 - 5 x + 17 x^{2} )( 1 + 5 x + 17 x^{2} )$
	$1 + 9 x^{2} + 289 x^{4}$
Frobenius angles:	$\pm0.292637436158$, $\pm0.707362563842$
Angle rank:	$1$ (numerical)
Jacobians:	$11$
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$299$	$89401$	$24130496$	$7059192361$	$2015996664539$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$18$	$308$	$4914$	$84516$	$1419858$	$24123422$	$410338674$	$6975597508$	$118587876498$	$2015999428628$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 11 curves (of which all are hyperelliptic):

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

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.af $\times$ 1.17.f 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.af : \(\Q(\sqrt{-43}) \).
• 1.17.f : \(\Q(\sqrt{-43}) \).

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

The base change of $A$ to $\F_{17^{2}}$ is 1.289.j^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-43}) \)$)$

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.ak_ch	$2$	(not in LMFDB)
2.17.k_ch	$2$	(not in LMFDB)
2.17.a_aj	$4$	(not in LMFDB)