Every residue.
One extra point.
Find the modular pairs that satisfy an elliptic equation. Then inspect when its elliptic interpretation stops applying.
Count dial0–64; trace dial−1…1 after dividing by2√p. Bright points are solutions; ringed points share the selected x.
Reduced motion is enabled. Inspect still residue columns with the controls.
What this finite count means
y²=x³+ax+b modulo p, with prime p>3. Coordinates are residues0…p−1, not real points joined by a continuous curve. Bright dots mark every exact affine solution; a highlighted column has0,1 or2 roots. Nonzero y pairs with p−y; y0 counts once.
One projective point O=[0:1:0] is added beyond the grid, even in a singular case. N=affine count+1. Changing p changes the labelled residue grid and clamps a,b,x; all dials keep fixed scales. Manual sweeps stop at the last column.
Δ=−16(4a³+27b²) mod p. Only Δ≠0 gives an elliptic curve, its Frobenius trace t=p+1−N and Hasse comparison |t|≤2√p. If Δ=0, the trace readout is N/A and its muted parked needle does not assign zero. The separate arithmetic difference remains available.
Finite reduction counts feed local L-function factors at good primes. They do not determine rational rank, analytic continuation, L(E,1), Sha, BSD, Hodge classes or complex geometry. This independently authored teaching counter does not verify a paper, Lean proof or the episode's research claims.
MIT18.783: point counting, quadratic residues and Hasse's bound ↗
MIT18.783: discriminant and local trace context ↗