There are 11 negative integers D such that there is a quadratic order
of discriminant D with class number 1 and no roots of unity other than
1 and -1. For each of those there is a unique K3 surface X/Q
with Neron-Severi group of rank 20 and discriminant -D that consists
entirely of classes of divisors defined over Q. These surfaces
can be constructed uniformly from the Weierstrass equation of a
CM elliptic curve of discriminant -D, via the Shioda-Inose correspondence;
but for large |D| that forces at least one of the generators of NS(X)
to be very complicated! For each of these surfaces we give an explicit
model as an elliptic fibration with E8(=II*) fibers at t=0 and t=infinity,
and an A1(=I2) fiber at t=-1. The components of these fibers, together with
the zero-section of the fibration, generate a subgroup of NS(X) of rank 19
and discriminant 2. The last generator is a section of canonical height |D|/2
that is also a generator of the infinite cyclic Mordell-Weil group of
sections of the fibration. We list these generators in each case,
giving only the x-coordinate once |D|>19 because even the formulas for
x as a rational function of t are getting enormous -- especially in the
final case of D = -163 where the coefficients of the numerator of x
already have more than 104 digits altogether!
D = -7: y^2 = x^3 - 75*x - (64*t + 378 + 64/t)
D = -8: y^2 = x^3 - 675*x + 27*(27*t - 196 + 27/t)
D = -11: y^2 = x^3 - 1728*x - 27*(27*t + 1078 + 27/t)
D = -12: y^2 = x^3 - 75*x + (4*t - 242 + 4/t)
D = -16: y^2 = x^3 - 363*x + (8*t - 2646 + 8/t)
D = -19: y^2 = x^3 - 192*x - (t + 1026 + 1/t)
D = -27: y^2 = x^3 - 4800*x - (9*t + 128018 + 9/t)
D = -28: y^2 = x^3 - 21675*x + (64*t - 1228122 + 64/t)
D = -43: y^2 = x^3 - 19200*x - (t + 1024002 + 1/t)
D = -67: y^2 = x^3 - 580800*x - (t + 170368002 + 1/t)
D = -163: y^2 = x^3 - 8541868800*x - (t + 303862746112002 + 1/t)
For the two cases D = -3 and D = -4 where the quadratic order has
class number 1 but extra roots of unity, there is still a surface X/Q with
rank(NS(X))=20, disc(NS(X))=D, and NS(X)=NSQ(X); but X is unique only over C,
not over Q where there are respectively cubic and quadratic twists.
X still has an elliptic fibration with E8(=II*) fibers at t=0 and t=infinity
and a reducible fiber at t=-1; but for D = -3 the fiber at t=-1 contributes
A2 to NS(X) (with Kodaira type IV, not the usual I3), while for D = -4
there is a second A1=I2 fiber at t=+1. In both cases the Mordell-Weil
group is trivial. The Weierstrass models are
y^2 = x^3 + c^2*(t + 2 + 1/t)
for D = -3 (with nonzero c determined mod Q*3), and
y^2 = x^3 - 3*c^2*x + c^3*(t+1/t)
for D = -4 (with nonzero c determined mod Q*2).
======================================================================
D = -7:
y^2 = x^3 - 75*x - (64*t + 378 + 64/t)
MW generator of height 7/2:
x = (4*t^4+36*t^3+19*t^2+36*t+4) / (3*t)^2
y = 4 * (t^2-1) * (2*t^4+27*t^3+77*t^2+27*t+2) / (3*t)^3
======================================================================
D = -8:
y^2 = x^3 - 675*x + 27*(27*t - 196 + 27/t)
MW generator of height 4:
x = 3 * (3*t^4-48*t^3+58*t^2-48*t+3) / (4*t)^2
y = 27 * (t^2-1) * (t^4-24*t^3+126*t^2-24*t+1) / (4*t)^3
[NB no height correction at the I_2 fiber t=-1, even though y vanishes
there, because x(-1) = 30 and the non-identity component has x(-1) = -15]
======================================================================
D = -11:
y^2 = x^3 - 1728*x - 27*(27*t + 1078 + 27/t)
MW generator of height 11/2:
x = 3 * (3*t^6+210*t^5+301*t^4+8380*t^3+301*t^2+210*t+3) / (16*(t^2-t))^2
{
y = 27 * (t+1) * (
t^8+104*t^7+1884*t^6-13864*t^5-64058*t^4-13864*t^3+1884*t^2+104*t+1
) / (16*(t^2-t))^3
}
======================================================================
D = -12:
y^2 = x^3 - 75*x + (4*t - 242 + 4/t)
MW generator of height 6:
x = (t^6-114*t^5+555*t^4-8660*t^3+555*t^2-114*t+1) / (18*(t^2+t))^2
{
y = (t-1) * (
t^8-170*t^7+5536*t^6+37690*t^5-436034*t^4+37690*t^3+5536*t^2-170*t+1
) / (18*(t^2+t))^3
}
======================================================================
D = -16:
y^2 = x^3 - 363*x + (8*t - 2646 + 8/t)
MW generator of height 8 has (x,y) = (X_NUM / DEN^2, Y_NUM / DEN^3) where
{
X_NUM =
t^8
- 608*t^7
+ 12568*t^6
- 1441120*t^5
- 599122*t^4
- 1441120*t^3
+ 12568*t^2
- 608*t
+ 1;
Y_NUM = (t^2-1) * (
t^10
- 912*t^9
+ 157477*t^8
+ 6153632*t^7
- 396616406*t^6
+ 1898748512*t^5
- 396616406*t^4
+ 6153632*t^3
+ 157477*t^2
- 912*t
+ 1);
DEN = 6 * t * (11*t^2 + 76*t + 11);
}
[NB as with D=-8, there is no height correction at the I_2 fiber t=-1,
even though y vanishes there, because x(-1) = 22 and the non-identity
component of the I_2 fiber has x(-1) = -11.]
======================================================================
D = -19:
y^2 = x^3 - 192*x - (t + 1026 + 1/t);
MW generator of height 19/2 has (x,y) = (X_NUM / DEN^2, Y_NUM / DEN^3) where
{
X_NUM =
t^10
+ 1866*t^9
+ 99373*t^8
+ 43378552*t^7
- 166040366*t^6
+ 789316732*t^5
- 166040366*t^4
+ 43378552*t^3
+ 99373*t^2
+ 1866*t
+ 1;
Y_NUM = (t+1) * (
t^14
+ 2798*t^13
+ 1451995*t^12
- 203394772*t^11
- 34859885207*t^10
- 371855939854*t^9
+ 1157876288187*t^8
+ 4845580245480*t^7
+ 1157876288187*t^6
- 371855939854*t^5
- 34859885207*t^4
- 203394772*t^3
+ 1451995*t^2
+ 2798*t
+ 1);
DEN = 48 * (t^2-t) * (5*t^2 - 118*t + 5);
}
======================================================================
D = -27:
y^2 = x^3 - 4800*x - (9*t + 128018 + 9/t)
MW generator of height 27/2 has x = X_NUM / DEN^2 where
{
X_NUM =
t^14
+ 25934*t^13
+ 20056411*t^12
+ 115235119084*t^11
- 5459534149399*t^10
+ 355219952434834*t^9
- 1533931033920069*t^8
+ 4225988295301224*t^7
- 1533931033920069*t^6
+ 355219952434834*t^5
- 5459534149399*t^4
+ 115235119084*t^3
+ 20056411*t^2
+ 25934*t
+ 1;
DEN = 48 * (t^2-t) * (31*t^4 - 10244*t^3 + 126906*t^2 - 10244*t + 31);
}
======================================================================
D = -28:
y^2 = x^3 - 21675*x + (64*t - 1228122 + 64/t)
MW generator of height 14 has x = X_NUM / DEN^2 where
{
X_NUM =
4*t^14
- 140024*t^13
+ 147972784*t^12
- 1131773297524*t^11
- 66285653689621*t^10
- 6004157195188492*t^9
- 26267927054746548*t^8
- 91635020035237182*t^7
- 26267927054746548*t^6
- 6004157195188492*t^5
- 66285653689621*t^4
- 1131773297524*t^3
+ 147972784*t^2
- 140024*t
+ 4;
DEN = 81 * (t^2+t) * (34*t^4 + 15011*t^3 + 233079*t^2 + 15011*t + 34);
}
======================================================================
D = -43:
y^2 = x^3 - 19200*x - (t + 1024002 + 1/t);
MW generator of height 43/2 has x = X_NUM / DEN^2 where
{
X_NUM =
t^22
+ 1867542*t^21
+ 104677627111*t^20
+ 42991700221106180*t^19
- 140770990749704302445*t^18
+ 668469030837589556820446*t^17
- 191021297358312923276135563*t^16
+ 36660997743169958190035890736*t^15
- 1104340739099409471012128837350*t^14
+ 17439658441545278921193839199500*t^13
- 67819854972038604255137999915530*t^12
+ 126711085729291662443215827103640*t^11
- 67819854972038604255137999915530*t^10
+ 17439658441545278921193839199500*t^9
- 1104340739099409471012128837350*t^8
+ 36660997743169958190035890736*t^7
- 191021297358312923276135563*t^6
+ 668469030837589556820446*t^5
- 140770990749704302445*t^4
+ 42991700221106180*t^3
+ 104677627111*t^2
+ 1867542*t
+ 1;
DEN =
288 * (t^2-t) * (
263*t^8
- 6241960*t^7
+ 5772878468*t^6
- 391530832280*t^5
+ 2564612094058*t^4
- 391530832280*t^3
+ 5772878468*t^2
- 6241960*t
+ 263
);
}
======================================================================
D = -67:
y^2 = x^3 - 580800*x - (t + 170368002 + 1/t);
MW generator of height 67/2 has x = X_NUM / DEN^2 where
{
X_NUM =
t^34
+ 310713954*t^33
+ 2898107609183281*t^32
+ 197992809952986470743904*t^31
- 107672600771016377937170552024*t^30
+ 85124790521113340972706563615311088*t^29
- 4010696694433718974715684106523655958648*t^28
+ 127834651462180912014204572857317005136784928*t^27
- 621222041513358504953391322500286788308172113748*t^26
+ 1606896258113561016672387715368812812816723911202488*t^25
- 988859446692184873658765627059999418550984430907914932*t^24
+ 298055597816168934851446441230366075486466592743884523872*t^23
- 27697249841371617156039441752918845979158361739474040605192*t^22
+ 1233928852273342620308030535103565883835110425043122935485072*t^21
- 20060775799983990139351607902170503888628677399452312939995432*t^20
+ 156929477283786467471394799341031848743981427559860736697370912*t^19
- 498243673584100885548584116363549473971104026830692597958885802*t^18
+ 767277271950987278886115271512654492413227975936933279602042572*t^17
- 498243673584100885548584116363549473971104026830692597958885802*t^16
+ 156929477283786467471394799341031848743981427559860736697370912*t^15
- 20060775799983990139351607902170503888628677399452312939995432*t^14
+ 1233928852273342620308030535103565883835110425043122935485072*t^13
- 27697249841371617156039441752918845979158361739474040605192*t^12
+ 298055597816168934851446441230366075486466592743884523872*t^11
- 988859446692184873658765627059999418550984430907914932*t^10
+ 1606896258113561016672387715368812812816723911202488*t^9
- 621222041513358504953391322500286788308172113748*t^8
+ 127834651462180912014204572857317005136784928*t^7
- 4010696694433718974715684106523655958648*t^6
+ 85124790521113340972706563615311088*t^5
- 107672600771016377937170552024*t^4
+ 197992809952986470743904*t^3
+ 2898107609183281*t^2
+ 310713954*t
+ 1;
DEN =
48 * (t^2-t) * (
111947*t^14
- 441985046758*t^13
+ 67936338448223657*t^12
- 767504422069507489628*t^11
+ 914545853627528982105667*t^10
- 146424712594682169094539386*t^9
+ 3705842056991910077990297289*t^8
- 15160041717878842747984371336*t^7
+ 3705842056991910077990297289*t^6
- 146424712594682169094539386*t^5
+ 914545853627528982105667*t^4
- 767504422069507489628*t^3
+ 67936338448223657*t^2
- 441985046758*t
+ 111947
);
}
======================================================================
D = -163:
y^2 = x^3 - 8541868800*x - (t + 303862746112002 + 1/t);
MW generator of height 163/2 has x = X_NUM / DEN^2 where
{
X_NUM = t^82
+ 554179195815762*t^81
+ 9219222870014741288220501241*t^80
+ 1123356913598880042682576528762746769895920*t^79
- 1089563951614958004708392358458598702959494440288667580*t^78
+ 1536373813544370367872253742655159940110934679832826683280163507032*t^77
- 129086290343681320425049623890281724429780794530271014252121743337451583577196*t^76
+ 7338013044736677938228381003301335451687137995941361953458453379719257191559495013260432*t^75
- 63545913982909225500316035735083999806894149200681748651632306036835711578900305740982430135241530*t^74
+ 292993112298485391579628187119179855466006481786913832887482426082048098466460563765016228483560056319357020*t^73
- 320421776229995488877560281699417960747926537810130685296586021199313992325829909400033225094266032677813720800150522*t^72
+ 171690550983450674972179579498133608818730219197573396846255983419574281155485469023587104032138847355593927013579908707511056*t^71
- 28131781939014058670503643490136821886688315217454678833151760118910576171688290124810224285515051390843926791677405545568579330056332*t^70
+ 2210125895363019995160989297054328584851379333139146938074561730145390156729611936648873377525068218540888174367489280358205600984374987366040*t^69
- 62436983751796851740743200779010399045756401261085316650567465776958289627716361928803856090389031067051603688154472213303829921254400576746051872860*t^68
+ 849874976724778023046623220145257952312497403313054600727968770755423165710304045327782551188944187330560915654184430525364532373767738882486205420372436848*t^67
- 4649829146943372990025616762495347153581487073979061551058703341178138765873647363182802457688762311599137108747281836487013422729518712553014367228381541386855259*t^66
+ 12461287746958455208977880151992471280570758034580624916257225781197381603141947001934642362964466802477153489009642246029307841334640895199995807152460507618138853748458*t^65
- 14556274726559452824341916040457361501962354369873924674197810668238606644729660573484005375623776827783248879035061908062117467361987239986781589035571610008624020437778575155*t^64
+ 8507399559761974503653158343615196899400108789784321943847932646073519477191322569773578224606552724418882878946912593624250921125581132728192108176754785400869116177887415424272320*t^63
- 2305202517612954262783385328430143367969269740902084066101114601350614505017463301766866760650256018027280465602852516634917269668954656840889106174144343663399713393312070689644406798640*t^62
+ 319691223169788678667195655544846872555107408299722001583703859967886209056399870107668389582819796198503233487211350660591001026614530875521994480096057498705252653619207453908245812731811040*t^61
- 21582654120979268864816219868153702059951604040813054829389550180711595087262094187823952303957105202333775353815507143478015881169063609465435034628237642541535918697005400181106510057374048756720*t^60
+ 762756480316958238399870428358822724243901177270458541918560808146177594043815931427885390909870840236423897356195026809958082164300141710347599632236222419917138695891016023013156594191471108340630080*t^59
- 13654848783297547238101294859432000055049296087209319759066425924879151303412731234964535921407787315397286101910915495974263283309347116586217097636135403822784942407406496381366870445409473802094316172920*t^58
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- 655869887310230315515170380394148424848671635291859341401085957017698861849338058211309955302182751069374299748808191431948583580214284106679674936717157845445775668055414681159304654577683858159838404462607905528*t^56
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- 2650690398450118741075657626183896304404036245173542034366586375047896717095108315409618689887005649727249606516476215711534043083767828984997859036159505314793410558821008077704253310354994077961733402739758637064592560*t^54
+ 2177796804736359330042612924011807736904090904605684048730423250113815868437726649747924281648440839337782466733901280294319155369544927841422711644179690852719478383695163472489873847930286771693812782471151274651054905440*t^53
- 988337338747920724132383505385646982151372436877792445490516911950533469410960367528425042696439018180591827062774080383239226623466814207481866018415800419736029549644809531491414088720146136704873725972425854504930700684528*t^52
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- 36962872368968546565939375907352821367888848830244004138745017048789671085615937988835096468552127826977735116947039551981548515770137648240479444737799308096805561793042475198137436208058502780582492153783608557706606400474460558*t^50
+ 3099078246586133407360047268615956693444041179272524903011623359100396744043863031878437380877331719844110449466862509378835593980937478566235117928127816187206987350105465292667030948556118686429813228020505343097172963456806901700*t^49
- 149709630347662954692349673042573291919637892964180186124072549311441233315024831987047615289373216461393109832290384127206968294219341521635941309140663341101967702556213639238200808366013965830290017094082191037205583445424275286750*t^48
+ 4240096007719845804151587767431925532953062502019335921807348528791106183254001858418063111989261369147417458658221315643954522942664287244486597271523985483459976161725511497357950865818818787348340265198815454390350854722599705458208*t^47
- 70433963188642912458444262610541281068642652675236938517714981624944843731114968003677512014994254171991062314501864106403539726243487885063973133052421599520271615328411622409195846778519170723804818000741826606241071678696386018972104*t^46
+ 695883026600094971855764591543842772600474349862753799486467369144189310325503810954700486440760405940182721697918185895205833767155142157300558348991003940415574076037922695530703410627116731762470791900067868924608290975072219939154128*t^45
- 4092396991852571713468346626847697960684474224837177036820699546218281591768716573770257280623921634885713532781997008335639913538827555221237697768751628644116805271172246596195999203251958596470086280017111180671849433999191307578088040*t^44
+ 14475962743355478978626392477179280539165179802246801317135843168442580804311814776745740181302891113663400236826093231471985206238779116001090381867499813781397403141428630017208765983609385449000645110883177111467685905713108049791220960*t^43
- 30787442443050291273712732327987488286835426813370143105331706122424192032206174122815828491765006966597034798678492238605378369496238194432763834238940200734348214422955836584076507407332369389454524346808694476853927152656184836400987484*t^42
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}
{
DEN = 96 * (t^2-t) * (
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);
}