Abelian variety isogeny class 3.19.av_hs_abqs over $\F_{19}$

• Label: 3.19.av_hs_abqs
• Base field: $\F_{19}$
• 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:	$\F_{19}$
Dimension:	$3$
L-polynomial:	$( 1 - 8 x + 19 x^{2} )( 1 - 13 x + 77 x^{2} - 247 x^{3} + 361 x^{4} )$
	$1 - 21 x + 200 x^{2} - 1110 x^{3} + 3800 x^{4} - 7581 x^{5} + 6859 x^{6}$
Frobenius angles:	$\pm0.0986133210333$, $\pm0.130073469147$, $\pm0.318874605641$
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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$2148$	$42040656$	$323116768548$	$2218091412091968$	$15183864271616319168$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-1$	$321$	$6869$	$130601$	$2476544$	$47047149$	$893918255$	$16984068545$	$322690782881$	$6131078213676$

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 $\F_{19}$.

Endomorphism algebra over $\F_{19}$

The isogeny class factors as 1.19.ai $\times$ 2.19.an_cz and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.19.ai : \(\Q(\sqrt{-3}) \).
• 2.19.an_cz : 4.0.64389.1.

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
3.19.af_ai_es	$2$	(not in LMFDB)
3.19.f_ai_aes	$2$	(not in LMFDB)
3.19.v_hs_bqs	$2$	(not in LMFDB)
3.19.am_df_aqb	$3$	(not in LMFDB)
3.19.ag_f_bt	$3$	(not in LMFDB)