Let us consider the elliptic curve $C$ over $ℚ$ in Weierstrass form
$$C:y²=x³+1$$
How I can calculate the algebraic rank of $C(ℚ)$.
2026-03-29 15:40:31.1774798831
How I can calculate the algebraic rank of $C(ℚ)$
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The rank of this curve is $0$. Extensive computations for all curves $y^2=x^3+k$ in the range $|k| \leq10^5$ was made by Gebel, Pethö and Zimmer in their paper On Mordell's Equation which you may find here
http://www.inf.unideb.hu/~pethoe/cikkek/67_MORDELL.pdf
The table of their results may be resurrected here
http://web.archive.org/web/20040816044321/http://emmy.math.uni-sb.de/~simath/MORDELL/MORDELL+