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

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

Invariants

Base field:	$\F_{17}$
Dimension:	$2$
L-polynomial:	$1 + 4 x - x^{2} + 68 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.327873003470$, $\pm0.994539670137$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\sqrt{-3}, \sqrt{-13})\)
Galois group:	$C_2^2$
Jacobians:	$0$
Isomorphism classes:	4

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

Newton polygon

This isogeny class is ordinary.

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

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$361$	$78337$	$25542916$	$6954523849$	$2017931027401$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$22$	$272$	$5194$	$83268$	$1421222$	$24118022$	$410354582$	$6975654916$	$118589237578$	$2015992921232$

Jacobians and polarizations

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

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{17^{3}}$.

Endomorphism algebra over $\F_{17}$

The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-13})\).

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

The base change of $A$ to $\F_{17^{3}}$ is 1.4913.fk^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-13}) \)$)$

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.ae_ab	$2$	(not in LMFDB)
2.17.ai_by	$3$	(not in LMFDB)
2.17.a_s	$6$	(not in LMFDB)