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

• Label: 2.17.aj_cc
• 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: no

Invariants

Base field:	$\F_{17}$
Dimension:	$2$
L-polynomial:	$( 1 - 5 x + 17 x^{2} )( 1 - 4 x + 17 x^{2} )$
	$1 - 9 x + 54 x^{2} - 153 x^{3} + 289 x^{4}$
Frobenius angles:	$\pm0.292637436158$, $\pm0.338793663197$
Angle rank:	$2$ (numerical)
Jacobians:	$0$
Isomorphism classes:	2

This isogeny class is not 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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$182$	$92092$	$25492376$	$7038775744$	$2014450139702$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$9$	$317$	$5184$	$84273$	$1418769$	$24120722$	$410282721$	$6975780001$	$118588837968$	$2015997643757$

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 $\F_{17}$.

Endomorphism algebra over $\F_{17}$

The isogeny class factors as 1.17.af $\times$ 1.17.ae 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.af : \(\Q(\sqrt{-43}) \).
• 1.17.ae : \(\Q(\sqrt{-13}) \).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.17.ab_o	$2$	(not in LMFDB)
2.17.b_o	$2$	(not in LMFDB)
2.17.j_cc	$2$	(not in LMFDB)