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

• Label: 2.17.a_bi
• Base field: $\F_{17}$
• Dimension: $2$
• $p$-rank: $0$
• Ordinary: no
• Supersingular: yes
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{17}$
Dimension:	$2$
L-polynomial:	$( 1 + 17 x^{2} )^{2}$
	$1 + 34 x^{2} + 289 x^{4}$
Frobenius angles:	$\pm0.5$, $\pm0.5$
Angle rank:	$0$ (numerical)
Jacobians:	$14$
Cyclic group of points:	no
Non-cyclic primes:	$2, 3$

This isogeny class is not simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is supersingular.

$p$-rank:	$0$
Slopes:	$[1/2, 1/2, 1/2, 1/2]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$324$	$104976$	$24147396$	$6879707136$	$2015996740164$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$18$	$358$	$4914$	$82366$	$1419858$	$24157222$	$410338674$	$6975423358$	$118587876498$	$2015999579878$

Jacobians and polarizations

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

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

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.a^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-17}) \)$)$

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

The base change of $A$ to $\F_{17^{2}}$ is 1.289.bi^2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $17$ and $\infty$.

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.a_ar	$3$	(not in LMFDB)
2.17.a_abi	$4$	(not in LMFDB)
2.17.a_a	$8$	(not in LMFDB)