Tables of fundamental integer solutions (x,y,z) of x3 + y3 = p2 z3
with p a prime congruent to 4 mod 9 and less than 1000
or congruent to 7 mod 9 and less than 666

See the page concerning E(p): x3 + y3 = p z3 for the background and notations. Essentially the same construction I use to find a nontorsion point on the curve E(p), for p a prime congruent to 4 or 7 mod 9, works for E(p2) as well. This time the heights of the points should grow as p2/3 rather than p1/3, which makes the computing time p7/3+o(1) (or p5/3+o(1) with convolution tricks). This is larger than the estimate for E(p), but smaller than the E(p') estimate for p' a prime large enough to get comparable height (i.e., for p' of size roughly p2). It is also smaller by a factor of p than the work required for the standard Heegner-point method; this factor is the same as for the curves E(p), and occurs for the same reason: only p or so terms of the q-expansion are needed, rather than p2.

This time it's the 9k+7 case that yields points of larger height by a factor of about 2, again due to the Tamagawa number at 3; and the 9k+4 case for which I get points that ``mysteriously'' are always divisible by 2. Additional divisibility by 2, which presumably detects |SHA|=4, occurs for p=331 and 367 in the p=9k+7 table, and for p=229 and 733 in the p=9k+4 table. Unlike the E(p) case, I was not able to confirm in all cases that SHA[2] has the correct order; it took MWRANK almost a week on a 500MHz machine just to check that each of E(3312) and E(3672) has |SHA[2]|=4 while E(p2) has trivial SHA[2] for each of the remaining eleven primes p=9k+7<400. The ``Caveat'' concerning the divisibility of points listed in the E(p) tables applies here as well.


p=9k+4

p (S/2)*g
13 (8, -7, 1)
31 (59, 13, 6)
67 (778, -653, 35)
103 (2166723410, 1421880881, 107137849)
139 (2909512930, -2905075579, 17991381)
157 (27242293, 8256515, 944682)
193 (28193071, -27488102, 353063)
211 (9107352571, 19190624054, 560103195)
229 2*(154699, -69515, 4004)
283 (517, 59, 12)
337 (7773971682119096, -5751462787384249, 135022593649447)
373 (1327359337099568639734706, 619668284602127931691697, 26457143903003388888481)
409 (359955858924981535914107, -217185492476356489389778,6014445413171715190771)
463 (5978078190808241541830, -1528169552414064089881, 99327377290254317071)
499 (2715774406168951739408421533826844293310052419, -2688433324849130766608922172846795406365164715, 13398243846501821193627569898421201168021794)
571 (39732784786662392932, -21208079785871321585, 546400129462338647)
607 (8269526905915553388003309040952667290663, 3647253707707579864369905386210563512470, 118559763325041838397582280823570307887)
661 (36967967758840, -8952988044631, 484864299009)
643 (11044437188815344938425994950492060849, -9641915624017731752674492411671805561, 102926503496839786563358956052269018)
733 2*(13143563507087, 2190757896497, 161924521444)
751 (141457238649858706730438807503308433463947216415474058774631433, 8698593413217318302806518544239731879898491810799946628121916, 1712243563864315412347798637037531945421646567225887454873577)
769 (4650867193435037960702968488352174747838902, -2498525922784285310175055385661205571502593, 52383525991920431522105385475599439253831)
787 (4424540873379809135908, -4391878835118173576585, 14540539730307768047)
823 (2771534574168782427911625688690837936459907764, -2661209531265267415501351070668953213288360091, 15334794628454190313307027041799429575977333)
859 (539869795937332902014291761130132116201942202573910033165176630646598934190954902363185070, -144504493848116066578638350290251333405521361778265667376660501029388316502384944351878897, 5935953534701742868041477614680498816477130798896412889570187512010296938832936652341823)
877 (1072108511358860782188562147693857690696412745754754303441892801096683102, -1025165666684747287756445121434690277032084990302869217024622972908187825, 5861446515689308665018421296950472678578105145795259503502544131534703)
967 (23199323930560569902599117877053552519214478645050, -4619927812994580454824362182479966350361024472333, 236615516516536613327257724298642128622272809723)

p=9k+7

p S*g
7 (11, -2, 3)
43 (7895419, -420163, 643242)
61 (93798938763304, -23969161192147, 6019245244341)
79 (1418322935604634846, 416502767358398513, 77680272383924217)
97 (440320075625234, 76769228526793, 20893884519009)
151 (74646062310742, -46249459600075, 2404562995917)
223 (4185244080304557503874779, 3852503919925757453019373, 137928337802602601406054)
241 (21697421209091419261966, 17446866263750020316159, 644173253960422059975)
277 (853951722249887705, -121817790815902049, 20076713596822314)
313 (355507307842882624593086325021133856149447336710120844428552934573043094018915289363, -354602746692986709129018423204648314355484458881941451025238387384142099383045862152, 1517122651849438712721950935044230084378368307868200665761294465082177989014675811)
331 2*(4005903095210, 5002199335033, 120026907273)
349 (4283230035127376745758371723, 10504911120076144283768847290, 216605813855001873448924923)
367 2*(20378, -13817, 351)
421 (1269879740967134445894062774892534594766759, 1842305270194980826286501462089129022926568, 36045497524397859562891166918897597014191)
439 (55724937144157871697336013900472125169, 91957369671300126895090479002997888847, 1702276367697701052492492274210388748)
457 (140833139978792757081148437355901500373464447151775262501, -126029027926209685368581702509352164966560781039671262501, 1559110302812528160353633237483376331796867180536774200)
547 (28976867624335432403296963333080895955732630362589575771517260760283670, -28914518675909818098202463621800679083313651144484683488559901028751473, 80608579923312562847773397973189048573671668551484586185826396071983)
601 (5467278861476783908084563091, -1921703419448111081649556555, 75641680506277613325274566)
619 (1284776284600277115459286046554589032814911986898676, 597067111696317066359887491922567826012108387328115, 18261907644258104994391786814207411992702040395131)