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

• Label: 2.17.ac_an
• 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:	$1 - 2 x - 13 x^{2} - 34 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.0886875362893$, $\pm0.755354202956$
Angle rank:	$1$ (numerical)
Number field:	\(\Q(\zeta_{12})\)
Galois group:	$C_2^2$
Jacobians:	$0$
Isomorphism classes:	15
Cyclic group of points:	yes

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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$241$	$75433$	$23232400$	$7002671689$	$2012814185521$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$16$	$260$	$4726$	$83844$	$1417616$	$24139550$	$410378768$	$6975694084$	$118588986262$	$2015996087300$

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 \(\Q(\zeta_{12})\).

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

The base change of $A$ to $\F_{17^{3}}$ is 1.4913.adq^2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$

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.c_an	$2$	(not in LMFDB)
2.17.e_bm	$3$	(not in LMFDB)
2.17.ai_bv	$4$	(not in LMFDB)