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

• Label: 2.17.a_bh
• 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:	$( 1 - x + 17 x^{2} )( 1 + x + 17 x^{2} )$
	$1 + 33 x^{2} + 289 x^{4}$
Frobenius angles:	$\pm0.461304015105$, $\pm0.538695984895$
Angle rank:	$1$ (numerical)
Jacobians:	$2$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$323$	$104329$	$24144896$	$6890826121$	$2015994887843$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$18$	$356$	$4914$	$82500$	$1419858$	$24152222$	$410338674$	$6975569284$	$118587876498$	$2015995875236$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):

• $y^2=11 x^6+x^5+7 x^4+11 x^3+7 x^2+x+11$
• $y^2=16 x^6+3 x^5+4 x^4+16 x^3+4 x^2+3 x+16$

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.ab $\times$ 1.17.b 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.ab : \(\Q(\sqrt{-67}) \).
• 1.17.b : \(\Q(\sqrt{-67}) \).

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

The base change of $A$ to $\F_{17^{2}}$ is 1.289.bh^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-67}) \)$)$

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.ac_bj	$2$	(not in LMFDB)
2.17.c_bj	$2$	(not in LMFDB)
2.17.a_abh	$4$	(not in LMFDB)