An interesting family of curves of genus 1
We study the family of elliptic curves y2=x3−t2x+1, both over ℚ(t) and over ℚ. In the former case, all integral solutions are determined; in the latter case, computation in the range 1≤t≤999 shows large ranks are common, giving a particularly simple example of curves which (admittedly over a small r...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2000-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171200002210 |
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| Summary: | We study the family of elliptic curves y2=x3−t2x+1, both over
ℚ(t) and over ℚ. In the former case, all
integral solutions are determined; in the latter case, computation
in the range 1≤t≤999 shows large ranks are common, giving
a particularly simple example of curves which (admittedly over a
small range) apparently contradict the once held belief that the
rank under specialization will tend to have minimal rank consistent
with the parity predicted by the Selmer conjecture. |
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| ISSN: | 0161-1712 1687-0425 |