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

• Label: 2.17.a_bc
• Base field: $\F_{17}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: yes
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{17}$
Dimension:	$2$
L-polynomial:	$1 + 28 x^{2} + 289 x^{4}$
Frobenius angles:	$\pm0.403999106167$, $\pm0.596000893833$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{6}, \sqrt{-62})\)
Galois group:	$C_2^2$
Jacobians:	$14$
Cyclic group of points:	yes

This isogeny class is simple but not geometrically 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$318$	$101124$	$24135246$	$6941555856$	$2015991083118$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$18$	$346$	$4914$	$83110$	$1419858$	$24132922$	$410338674$	$6976006654$	$118587876498$	$2015988265786$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):

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

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{17}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{6}, \sqrt{-62})\).

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

The base change of $A$ to $\F_{17^{2}}$ is 1.289.bc^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-93}) \)$)$

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.a_abc	$4$	(not in LMFDB)