Enter an elliptic curve and choose a rank. The simulator goes through the squarefree integers in order of and finds the first quadratic twist whose Mordell–Weil group has that rank. Everything is computed live in your browser with PARI/GP.
If , its quadratic twist by a squarefree integer is , or equivalently . The curves and become isomorphic over , yet their groups of rational points can look completely different. In particular, the rank can go up or down.
Goldfeld's conjecture predicts that, counted by , half of all twists have rank 0, half have rank 1, and ranks occur with density zero. The running tally below lets you watch this happen.
How ranks are certified. PARI's ellrank runs a 2-descent that gives
lower and upper bounds for the rank. When they agree, the rank is proven. Otherwise the simulator computes
the analytic rank. If that is 0 or 1, the rank equals it by Gross–Zagier–Kolyvagin; if it is
, equality assumes the Birch–Swinnerton-Dyer conjecture,
and the result is labelled as such.
Fill in integers or fractions such as -1/16. Blank boxes count as 0.
Five numbers [a1,a2,a3,a4,a6] for the general form, or two numbers [a,b] for
. Pasting EllipticCurve([0,0,1,-7,6]) works too.
Order: (by , positive first). The maximum is .
Since , the twists by and are isomorphic, so they always have the same rank.
| d | Conductor of | Rank | How |
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