Specific Elliptic Curves: Rank - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T21:10:55Z http://mathoverflow.net/feeds/question/87410 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/87410/specific-elliptic-curves-rank Specific Elliptic Curves: Rank bobuhito 2012-02-03T08:48:12Z 2012-03-22T06:02:37Z <p>Here's a challenge for elliptic curve descenders/programmers. It seems no public software or public tables can determine if the rank is zero for the following curves (over rational x,y):</p> <pre><code> y^2 = x^3 - 9122*x + 106889 y^2 = x^3 - x^2 - 42144*x + 66420 y^2 = x^3 - x^2 - 168615*x + 21827700 y^2 = x^3 - 210386*x + 32627329 </code></pre> <p>Can anybody definitely say if any of their ranks are zero?</p> <p>By the way, these arose from Heronian triangles for a given base and height, so there are equivalent quartic forms which might be easier to analyze...for example, the rank of the first curve above is zero iff there are no rational s,t solutions to this equation:</p> <pre><code> ( s^2 - t^2 )^2 = 25*( 2*( s^2 + t^2 ) - 509 ) </code></pre> <p>Also, I have tried both Magma and Sage. Sage seems to be better at determining the rank in about 20% of similar cases. For example, "y^2 = x^3 + x^2 - 58055*x + 4135350" has rank 0 according to Sage, but Magma only bounds the rank from 0 to 2 (limited to one minute). Anyway, these 4 cases are unsolved.</p> http://mathoverflow.net/questions/87410/specific-elliptic-curves-rank/87412#87412 Answer by François Brunault for Specific Elliptic Curves: Rank François Brunault 2012-02-03T09:18:35Z 2012-02-03T09:18:35Z <p>Your elliptic curves \$E\$ all (provably) satisfy \$L(E,1) \neq 0\$, so by Kolyvagin's theorem , they have rank \$0\$. You can prove that \$L(E,1) \neq 0\$ by using the command <em>ellanalyticrank</em> in Pari/GP (there are similar commands in Magma and Sage).</p> <p>By the way, your first elliptic curve has conductor 223960, so is likely to be soon in Cremona's tables : <a href="http://www.warwick.ac.uk/staff/J.E.Cremona/ftp/data/INDEX.html" rel="nofollow">http://www.warwick.ac.uk/staff/J.E.Cremona/ftp/data/INDEX.html</a></p> http://mathoverflow.net/questions/87410/specific-elliptic-curves-rank/87426#87426 Answer by cjsh716 for Specific Elliptic Curves: Rank cjsh716 2012-02-03T12:30:20Z 2012-02-03T12:30:20Z <p>MAGMA online: all RANK =0,</p> <p>but indeed MAGMA sometime rank is defferent from Cremona's tables ,I meet one times before</p> <p>Qx := PolynomialRing(Rationals());</p> <p>E00:=EllipticCurve(x^3 - 9122*x + 106889 ); E00; MordellWeilShaInformation(E00); DescentInformation(E00); Generators(E00) ; Q00, reps := IntegralPoints(E00); Q00; TorsionSubgroup(E00);</p> <p>Elliptic Curve defined by y^2 = x^3 - 9122*x + 106889 over Rational Field</p> <p>Torsion Subgroup = Z/4 Analytic rank = 0 ==> Rank(E) = 0</p> <p>[ 0, 0 ] [] []</p> <p>Torsion Subgroup = Z/4 Analytic rank = 0 ==> Rank(E) = 0</p> <p>[ 0, 0 ] [] [] Warning: rank computed (0) is only a lower bound (It may still be correct, though) [ (-32 : 605 : 1) ] [ (-32 : 605 : 1), (89 : 0 : 1) ] Abelian Group isomorphic to Z/4 Defined on 1 generator Relations: 4*\$.1 = 0</p> <p>=========================</p> <p>Qx := PolynomialRing(Rationals());</p> <p>E00:=EllipticCurve(x^3 - x^2 - 42144*x + 66420); E00; MordellWeilShaInformation(E00); DescentInformation(E00); Generators(E00) ; Q00, reps := IntegralPoints(E00); Q00; TorsionSubgroup(E00);</p> <p>Elliptic Curve defined by y^2 = x^3 - x^2 - 42144*x + 66420 over Rational Field</p> <p>Torsion Subgroup = Z/4 Analytic rank = 0 ==> Rank(E) = 0</p> <p>[ 0, 0 ] [] []</p> <p>Torsion Subgroup = Z/4 Analytic rank = 0 ==> Rank(E) = 0</p> <p>[ 0, 0 ] [] [] Warning: rank computed (0) is only a lower bound (It may still be correct, though) [ (-84 : 1734 : 1) ] [ (-84 : 1734 : 1), (205 : 0 : 1) ] Abelian Group isomorphic to Z/4 Defined on 1 generator Relations: 4*\$.1 = 0</p> <p>===============</p> <p>Qx := PolynomialRing(Rationals());</p> <p>E00:=EllipticCurve(x^3 - x^2 - 168615*x + 21827700 ); E00; MordellWeilShaInformation(E00); DescentInformation(E00); Generators(E00) ; Q00, reps := IntegralPoints(E00); Q00; TorsionSubgroup(E00);</p> <p>Elliptic Curve defined by y^2 = x^3 - x^2 - 168615*x + 21827700 over Rational Field</p> <p>Torsion Subgroup = Z/4 Analytic rank = 0 ==> Rank(E) = 0</p> <p>[ 0, 0 ] [] []</p> <p>Torsion Subgroup = Z/4 Analytic rank = 0 ==> Rank(E) = 0</p> <p>[ 0, 0 ] [] [] Warning: rank computed (0) is only a lower bound (It may still be correct, though) [ (-45 : -5415 : 1) ] [ (-45 : -5415 : 1), (316 : 0 : 1) ] Abelian Group isomorphic to Z/4 Defined on 1 generator Relations:</p> <h1> 4*\$.1 = 0</h1> <p>Qx := PolynomialRing(Rationals());</p> <p>E00:=EllipticCurve(x^3 - 210386*x + 32627329); E00; MordellWeilShaInformation(E00); DescentInformation(E00); Generators(E00) ; Q00, reps := IntegralPoints(E00); Q00; TorsionSubgroup(E00);</p> <p>Elliptic Curve defined by y^2 = x^3 - 210386*x + 32627329 over Rational Field</p> <p>Torsion Subgroup = Z/4 Analytic rank = 0 ==> Rank(E) = 0</p> <p>[ 0, 0 ] [] []</p> <p>Torsion Subgroup = Z/4 Analytic rank = 0 ==> Rank(E) = 0</p> <p>[ 0, 0 ] [] [] Warning: rank computed (0) is only a lower bound (It may still be correct, though) [ (-24 : -6137 : 1) ] [ (-24 : -6137 : 1), (337 : 0 : 1) ] Abelian Group isomorphic to Z/4 Defined on 1 generator Relations: 4*\$.1 = 0</p> http://mathoverflow.net/questions/87410/specific-elliptic-curves-rank/87438#87438 Answer by John Cremona for Specific Elliptic Curves: Rank John Cremona 2012-02-03T14:08:43Z 2012-02-03T14:08:43Z <p>In reply to François Brunault: indeed this curve is 223960i1 and was first computed by me 4 days ago.</p> http://mathoverflow.net/questions/87410/specific-elliptic-curves-rank/87474#87474 Answer by John Cremona for Specific Elliptic Curves: Rank John Cremona 2012-02-03T20:31:51Z 2012-02-03T20:31:51Z <p>Bob: although you say that Magma disagrees with my tables, none of your examples actually shows this. You only show that using the most obvious Magma command cannot prove that the rank is zero. As has been pointed out, one can prove that the ranks are zero by computing the analytic ranks and applying certain Theorems. Or, you just have to go beyond 2-descent. The reason why 2-descent is insufficient for these curves is that both the curve and the 2-isogenous curve have Sha of order4. But Magma can also do 3- and 4-descent! It just does not use them automatically. A 3-descent should be able to show that the rank is 0, by computing that the 3-Selmer group is trivial. I say "should" because the 3-descent algorithm (developed and implemented by me with Tom Fisher, Michael Stoll, Cathy O'Neil and Denis Simon) involves working in the 3-division field and so can be slow. (I am running ThreeDescent(E) on the first of your curves as I type). Alternatively you can get the everywhere locally soluble 2-coverings in Magma and then apply FourDescent() to those.</p> <p>So it is not true that existing descent techniques are insufficient for these curves, but the tools available need to be used with some understanding of how they work</p>