Can we find small elliptic curves of high rank?
Inspired by Dujella's rank tables, this site tracks elliptic curves E/ℚ of high Mordell–Weil rank relative to their size — measured by conductor, height, or discriminant.
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Plots
Each dot is a curve — click one for its witness, or click a rank on the axis to list the curves at that rank. The frontier is to the right and down: high rank, small size.
Natural log of the conductor N = ∏p pfp. Recorded when a submission supplies the primes of bad reduction.
Naive height = log max(|c4|3, |c6|2) of the global minimal model. Recorded for every curve.
Stable Faltings height (LMFDB normalization). Recorded for every curve.
Natural log of the absolute discriminant |Δ| of the global minimal model. Recorded for every curve.
Submit a curve
Give the Weierstrass coefficients and a set of independent rational points. Each point is checked to lie on the curve, and their Néron–Tate height-pairing matrix is checked to be positive definite — so the points are independent in E(ℚ), proving rank ≥ the number of points. Supplying the primes of bad reduction additionally records its conductor.