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

• Label: 2.17.a_abh
• 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: no

Invariants

Base field:	$\F_{17}$
Dimension:	$2$
L-polynomial:	$1 - 33 x^{2} + 289 x^{4}$
Frobenius angles:	$\pm0.0386959848954$, $\pm0.961304015105$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(i, \sqrt{67})\)
Galois group:	$C_2^2$
Jacobians:	$0$

This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable.

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$257$	$66049$	$24130244$	$6890826121$	$2015992913057$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$18$	$224$	$4914$	$82500$	$1419858$	$24122918$	$410338674$	$6975569284$	$118587876498$	$2015991925664$

Jacobians and polarizations

This isogeny class is principally polarizable, but does not contain a Jacobian.

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 \(\Q(i, \sqrt{67})\).

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

The base change of $A$ to $\F_{17^{2}}$ is 1.289.abh^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	$4$	(not in LMFDB)
2.17.a_bh	$4$	(not in LMFDB)
2.17.c_bj	$4$	(not in LMFDB)