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

• Label: 3.17.as_fy_abeq
• 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:	$( 1 - 8 x + 17 x^{2} )( 1 - 10 x + 57 x^{2} - 170 x^{3} + 289 x^{4} )$
	$1 - 18 x + 154 x^{2} - 796 x^{3} + 2618 x^{4} - 5202 x^{5} + 4913 x^{6}$
Frobenius angles:	$\pm0.0779791303774$, $\pm0.216316574334$, $\pm0.356804757520$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1670$	$22882340$	$120917839640$	$584975123283200$	$2861700787044693350$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$0$	$274$	$5010$	$83858$	$1419500$	$24133528$	$410342380$	$6975860002$	$118588147890$	$2015993010114$

Jacobians and polarizations

This isogeny class is principally polarizable, but it is unknown whether it contains 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.ai $\times$ 2.17.ak_cf 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_cf : 4.0.94784.1.

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
3.17.ac_ag_em	$2$	(not in LMFDB)
3.17.c_ag_aem	$2$	(not in LMFDB)
3.17.s_fy_beq	$2$	(not in LMFDB)