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

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

Invariants

Base field:	$\F_{17}$
Dimension:	$3$
L-polynomial:	$( 1 - 8 x + 17 x^{2} )( 1 - 6 x + 17 x^{2} )^{2}$
	$1 - 20 x + 183 x^{2} - 968 x^{3} + 3111 x^{4} - 5780 x^{5} + 4913 x^{6}$
Frobenius angles:	$\pm0.0779791303774$, $\pm0.240632536990$, $\pm0.240632536990$
Angle rank:	$2$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$1440$	$21565440$	$120442476960$	$588401816371200$	$2868518216538295200$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-2$	$256$	$4990$	$84348$	$1422878$	$24140032$	$410300126$	$6975495932$	$118587089470$	$2015994439936$

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.ai $\times$ 1.17.ag^2 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 : \(\Q(\sqrt{-1}) \).
• 1.17.ag^2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$

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.ai_p_q	$2$	(not in LMFDB)
3.17.ae_aj_fw	$2$	(not in LMFDB)
3.17.e_aj_afw	$2$	(not in LMFDB)
3.17.i_p_aq	$2$	(not in LMFDB)
3.17.u_hb_blg	$2$	(not in LMFDB)
3.17.ac_am_ca	$3$	(not in LMFDB)