Abelian variety isogeny class 3.17.as_fu_abdk over $\F_{17}$

• Label: 3.17.as_fu_abdk
• Base field: $\F_{17}$
• Dimension: $3$
• $p$-rank: $3$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes

Invariants

Base field:	$\F_{17}$
Dimension:	$3$
L-polynomial:	$( 1 - 8 x + 17 x^{2} )( 1 - 10 x + 53 x^{2} - 170 x^{3} + 289 x^{4} )$
	$1 - 18 x + 150 x^{2} - 764 x^{3} + 2550 x^{4} - 5202 x^{5} + 4913 x^{6}$
Frobenius angles:	$\pm0.0779791303774$, $\pm0.141075462221$, $\pm0.399907016694$
Angle rank:	$3$ (numerical)

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

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$3$
Slopes:	$[0, 0, 0, 1, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1630$	$22164740$	$118021603960$	$579964360019200$	$2858881220456399150$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$266$	$4890$	$83138$	$1418100$	$24142232$	$410417700$	$6976109762$	$118588529370$	$2015994033386$

Jacobians and polarizations

This isogeny class is principally polarizable and contains no Jacobian of a hyperelliptic curve, but it is unknown whether it contains a Jacobian of a non-hyperelliptic curve.

Decomposition and endomorphism algebra

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

Endomorphism algebra over $\F_{17}$

The isogeny class factors as 1.17.ai $\times$ 2.17.ak_cb 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.ai : \(\Q(\sqrt{-1}) \).
• 2.17.ak_cb : 4.0.442944.2.

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
3.17.ac_ak_dg	$2$	(not in LMFDB)
3.17.c_ak_adg	$2$	(not in LMFDB)
3.17.s_fu_bdk	$2$	(not in LMFDB)