Explicit Normal Form of the Singular Local Hecke Order

Let the three rational 3-adic branches be ordered as (V7, V1, V5). Then the singular order has the explicit normal form

O_sing ≅ { (x_7, x_1, x_5) in Z_3^3 : x_1 ≡ x_5 mod 3 }
     ≅ Z_3 × {(a,b) in Z_3 × Z_3 : a ≡ b mod 3}.

So the V7 branch is free, while only the (V1,V5) pair is glued.

Modulo 27

operator	V7 coordinate	V1 branch	V5 branch	pair congruent mod 3?
T_11	12	3	24	True
T_43	17	25	13	True
T_67	14	16	13	True
T_163	1	26	26	True

Modulo 81

operator	V7 coordinate	V1 branch	V5 branch	pair congruent mod 3?
T_11	66	60	48	True
T_43	44	31	34	True
T_67	68	58	25	True
T_163	1	80	80	True

Modulo 243

operator	V7 coordinate	V1 branch	V5 branch	pair congruent mod 3?
T_11	228	237	195	True
T_43	125	7	139	True
T_67	149	241	4	True
T_163	1	242	242	True

Modulo 729

operator	V7 coordinate	V1 branch	V5 branch	pair congruent mod 3?
T_11	714	480	438	True
T_43	368	250	625	True
T_67	392	241	490	True
T_163	1	728	728	True

Read

This is the cleanest explicit local model obtained so far.

• the split V7 branch contributes the free Z_3 coordinate;
• the singularity is entirely in the pair (V1,V5);
• the gluing condition is exactly first-order congruence modulo 3 on that pair.

So the local Hecke node at level 163 is an explicit congruence order in Z_3^3, with one free branch and one glued pair.