Abelian variety isogeny class 3.7.am_cr_aiy over $\F_{7}$

• Label: 3.7.am_cr_aiy
• Base field: $\F_{7}$
• 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:	$\F_{7}$
Dimension:	$3$
L-polynomial:	$( 1 - 4 x + 7 x^{2} )^{3}$
	$1 - 12 x + 69 x^{2} - 232 x^{3} + 483 x^{4} - 588 x^{5} + 343 x^{6}$
Frobenius angles:	$\pm0.227185525829$, $\pm0.227185525829$, $\pm0.227185525829$
Angle rank:	$1$ (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

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$64$	$110592$	$48228544$	$15550119936$	$4951246821184$

Point counts of the (virtual) curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$-4$	$44$	$404$	$2684$	$17516$	$118508$	$822020$	$5752700$	$40315868$	$282409004$

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 $\F_{7}$.

Endomorphism algebra over $\F_{7}$

The isogeny class factors as 1.7.ae^3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$

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
3.7.ae_f_i	$2$	(not in LMFDB)
3.7.e_f_ai	$2$	(not in LMFDB)
3.7.m_cr_iy	$2$	(not in LMFDB)
3.7.aj_bt_afm	$3$	(not in LMFDB)
3.7.ag_be_adk	$3$	(not in LMFDB)
3.7.ad_ad_bm	$3$	(not in LMFDB)
3.7.ad_y_abr	$3$	(not in LMFDB)
3.7.a_a_u	$3$	(not in LMFDB)
3.7.d_m_bv	$3$	(not in LMFDB)
3.7.g_g_aq	$3$	(not in LMFDB)
3.7.j_bk_dx	$3$	(not in LMFDB)
3.7.p_ds_mx	$3$	(not in LMFDB)