# An elliptic curve y2=x3-Dx of rank 13

## New records! rank 14, etc.

An elliptic curve is of the form
E(D): y2 = x3 - D x
if and only if it has j-invariant 1728=123. These are the quartic twists of the curve E(1) of conductor 32. Curves E(D) and E(D') are isomorphic if D/D' is a fourth power, and 2-isogenous if D'=-4D.

On April 16, 2001, I found a choice of D that yields a curve E(D) of rank 13.

This value is

D = 2827529113871322622866959217 = 32 11 37 41 73 151 269 509 673 1193 2969 5233.

The thirteen points with x-coordinates

-53023103285625, -52826063038023, -52264238470308, -29162438222148, -27920725708647, -23843884093425, -20893472520447, -19924665190425, -11023069423473, -9645751205607, -7293017369343, 303030650052889, 2659109867774031

are independent; a 2-descent confirms that the rank is at most 13, so E(D) has rank exactly 13 -- as also certified (again using 2-descent) by Cremona's MWRANK.

The curve E(D) has at least 35 pairs of nontorsion integral points: the 13 listed above, and also those with x equal

-6318738986727, 53366153545551, 53525266719879, 56075012802831, 57802481969281, 63837482449081, 98640809753871, 113391065792271, 149574797988231, 179618209033404, 273811757148601, 287616875150521, 539583622182759, 849900850021884, 1481739498859009, 2289898325488921, 2463952792028124, 3193703671713159, 5880362094050304, 10194232424354319, 15804515158586364, 23035968210396144.

This is an exhaustive list of nonzero x=m2n (with n necessarily a factor of D/3) such that x3-Dx is a square and m4n<1028.

UPDATE (MAY 2002):

Mark Watkins has found several D of smaller absolute value for which E(D) has rank 13, the current record being

D = - 90389647280869401176648335 = - 5 7 23 43 83 97 101 269 461 503 4013 12829;

and the first example of rank 14:

D = - 402599774387690701016910427272483 = 32 7 11 17 19 23 37 59 71 73 97 127 139 151 263 313 443 733.

The integral points on E(-90389647280869401176648335) with nonzero x=m2n such that m4n<1028 are the 30 pairs with x-coordinates

109267165625, 699012151740, 1422638553735, 3291776981199, 5654458993409, 7605531940860, 7907148183375, 9195899733104, 13086347270535, 22835291094000, 27155267727809, 35526638964665, 39954023283335, 48336648553881, 50787429472871, 57018266762240, 90808117179929, 106643430244071, 108435296625404, 112346755846535, 143188064521625, 185831493473276, 204292256759759, 259078097950785, 1336722973133660, 2890532213069760, 7945856693043815, 12720102807737871, 15369386678218496, 236640671439118265;

for the 2-isogenous E(4*90389647280869401176648335), there are 19 such x:

-18963057690864, -18484663920460, -17426647820736, -17388343366540, -16155476668035, -8778464206515, -6825084354115, -1457056473235, -917742452940, -574290377435, 25607034796176, 30862886807104, 36257785838260, 44024656537701, 78007337635984, 629574520712564, 1980450402426000, 2225123063376500, 10308554465411860.

I have yet to attempt such a computation for the rank-14 curve.