11
$\begingroup$

We are searching for the rank $6$ elliptic curves with the torsion subgroup $\mathbb{Z}/8\mathbb{Z}$ using the families similar to Allan MacLeod's as described in

A. J. MacLeod, A Simple Method for High-Rank Families of Elliptic Curves with Specified Torsion, arXiv, Number Theory [math.NT] (2014), arXiv:1410.1662v1

and came across a curve

[1,0,0,-885470498073167713002317184212739304,315787360224933400897171748517120652503410335938363968]

Magma Calculator, Magma V2.24-7, and mwrank (with $-b14$) return $4$ generators for this curve, e.g.:

SetClassGroupBounds("GRH");
E:=EllipticCurve([1,0,0,-885470498073167713002317184212739304,315787360224933400897171748517120652503410335938363968]);
MordellWeilShaInformation(E);

Using model [ 1, 0, 0, -885470498073167713002317184212739304, 315787360224933400897171748517120652503410335938363968 ]
Torsion Subgroup = Z/8
The 2-Selmer group has rank 6
New point of infinite order (x = 291590635879268240110317582512/87995882881)
New point of infinite order (x = -780802002478160103839088529264/1152485984521)
New point of infinite order (x = 526632508060857475043443664/1355049721)
New point of infinite order (x = 25006056691267829443814801295058/11996130112849)
After 2-descent:
    4 <= Rank(E) <= 5
    Sha(E)[2] <= (Z/2)^1
(Searched up to height 10000 on the 2-coverings.)

Both Magma and mwrank return $5$ for the upper bound on rank of the curve and all the related 2-isogenies ($\mathbb{Z}/4\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}$, and $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ curves below) using all their 2-torsion points:

E:=EllipticCurve([1,0,0,-885470498073167713002317184212739304,315787360224933400897171748517120652503410335938363968]);TwoPowerIsogenyDescentRankBound(E);print(" ");
E:=EllipticCurve([1,0,0,6205127968803058885033015264909364456,-3406906009463658062905156451961488907125318013801796912]);TwoPowerIsogenyDescentRankBound(E);print(" ");
E:=EllipticCurve([1,0,0,-1810312155105870840254262815766788584,-457633210238033749039428724550432643513221271485300416]);G,h:=TwoTorsionSubgroup(E);T:=[h(g):g in G];TwoPowerIsogenyDescentRankBound(E,T[2]);TwoPowerIsogenyDescentRankBound(E,T[3]);TwoPowerIsogenyDescentRankBound(E,T[4]);print(" ");
E:=EllipticCurve([1,0,0,-24623218791538050601572671001307730104,-47007276920642307027416764645331974815957764878067127376]);G,h:=TwoTorsionSubgroup(E);T:=[h(g):g in G];TwoPowerIsogenyDescentRankBound(E,T[2]);TwoPowerIsogenyDescentRankBound(E,T[3]);TwoPowerIsogenyDescentRankBound(E,T[4]);print(" ");

5 [ 3, 3, 3, 3, 3 ]
[ 4, 4, 4, 4, 4 ]

5 [ 7, 7, 7, 7, 7 ]
[ 0, 0, 0, 0, 0 ]

5 [ 4, 4, 4, 4, 4 ]
[ 3, 3, 3, 3, 3 ]
5 [ 0, 0, 0, 0, 0 ]
[ 7, 7, 7, 7, 7 ]
5 [ 0, 0, 0, 0, 0 ]
[ 7, 7, 7, 7, 7 ]

5 [ 0, 0, 0, 0, 0 ]
[ 11, 7, 7, 7, 7 ]
5 [ 0, 0, 0, 0, 0 ]
[ 9, 7, 7, 7, 7 ]
5 [ 7, 7, 7, 7, 7 ]
[ 0, 0, 0, 0, 0 ]

We tried implementing Jeremy Rouse's and Zev Klagsbrun's (1 and 2) approaches to employ $2$- and $4$-coverings, but could not find a rational point for the intersection of two quadrics. We used different seeds in Magma Calculator and a bound up to $3\times10^8$, e.g.:

SetSeed(1);
SetClassGroupBounds("GRH");
E := EllipticCurve([1,0,0,-885470498073167713002317184212739304,315787360224933400897171748517120652503410335938363968]);
P1 := E![86609552610643851795318648321889070494,113996061058226165174165649863299331386757385142,41549112479447364007];
P2 := E![72445938379247589758672,7465810298311395376082337884312,117649];
P3 := E![36603887306503756637336310506125004116,9298633886669630228894900541541155002386034788,53504769840978593717];
P4 := E![29317967055802030746557715723363588500973254368,22661934910027931993184701249992172746172677836253523176,25876196859106536169778781677];
twocovers := TwoDescent(E : RemoveTorsion := true, RemoveGens := {P1,P2,P3,P4}); twocovers;
fourcovers := FourDescent(twocovers[1] : RemoveTorsion := true, RemoveGensEC := {P1,P2,P3,P4}); fourcovers;
_,m := AssociatedEllipticCurve(fourcovers[1] : E := E);
pts := PointsQI(fourcovers[1], 3*10^8 : OnlyOne := true);
pts[1];
m(pts[1]);

[
    Hyperelliptic Curve defined by y^2 = 4401556531432641*x^4 +
        189490115768444190*x^3 + 5942548110808516561*x^2 -
        11718931267663428016*x + 9974519367938837056 over Rational Field
]
[
    Curve over Rational Field defined by
    93*$.1^2 + 505*$.1*$.2 + 715*$.1*$.3 + 2737*$.1*$.4 + 182*$.2^2 +
        916*$.2*$.3 + 1233*$.2*$.4 + 690*$.3^2 - 1248*$.3*$.4 - 3453*$.4^2,
    43659*$.1^2 - 100751*$.1*$.2 + 100262*$.1*$.3 + 267364*$.1*$.4 + 57470*$.2^2
        + 183818*$.2*$.3 - 93739*$.2*$.4 - 647944*$.3^2 + 666862*$.3*$.4 -
    133155*$.4^2
]

>> pts[1];
      ^
Runtime error in '[]': Illegal null sequence

Question. Considering parity, there should be one more generator on the curve. Is there a way to find it?

We would greatly appreciate any hint leading to the discovery of the extra generator.

If you can compute an extra generator, your name will be published at the bottom of the $\mathbb{Z}/8\mathbb{Z}$ rank 5 page.

$\endgroup$
7
$\begingroup$

By pts := PointsQI(fourcovers[1], 3*10^12 : OnlyOne := true); I found the fifth point with the first coordinate

184125172284095573254772251800166095132866268069994053071269775232967258156536458883390620097481326304558948736/556947891689070160911189283508448977897416986918076204554508778044285190076205830897454330201.  

A slightly smaller point has the first coordinate

31207315512700919990861160205048461703358374278620437702635535252739296354266705346529925962/4817243728090299166338796143704937948812120388662168566804089530091841. 
$\endgroup$
2
  • $\begingroup$ Out of curiosity, about how much computation time did that take? $\endgroup$ Feb 19 '21 at 17:09
  • 4
    $\begingroup$ 2.5 hours (8838 seconds) $\endgroup$
    – duje
    Feb 19 '21 at 18:25
5
$\begingroup$

Although Magma's MordellWeilShaInformation can't find the fifth generator for the $\mathbb{Z}/8\mathbb{Z}$ curve, it can find the last generator for an isogenous $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}$ curve by searching up to height $10^5$ on the $4$-coverings.

SetClassGroupBounds("GRH");
E24 := EllipticCurve([1, 0, 0, -1810312155105870840254262815766788584, -457633210238033749039428724550432643513221271485300416]);
MordellWeilShaInformation(E);
Using model [ 1, 0, 0, -1810312155105870840254262815766788584, -457633210238033749039428724550432643513221271485300416 ]
Torsion Subgroup = Z/2 x Z/4
The 2-Selmer group has rank 7
New point of infinite order (x = -443706527227992154791056873744/470078898129)
New point of infinite order (x = -14367138002890543477931052222544/12054179879569)
New point of infinite order (x = -269944456262034200432888/241081)
New point of infinite order (x = -630597721484060507464551524086/528521730025)
After 2-descent:
    4 <= Rank(E) <= 5
    Sha(E)[2] <= (Z/2)^1
(Searched up to height 10000 on the 2-coverings.)
New point of infinite order (x = 13457909758544525530407577328967629341502974335624109249/9109819423915178769710709651398624256)
After 4-descent:
    5 <= Rank(E) <= 5
    Sha(E)[4] is trivial
(Searched up to height 10^5 on the 4-coverings.)

Each found generator can be mapped back to the original $\mathbb{Z}/8\mathbb{Z}$ curve.

E24 := EllipticCurve([1, 0, 0, -1810312155105870840254262815766788584, -457633210238033749039428724550432643513221271485300416]);
P1_24 := Points(E24, -443706527227992154791056873744/470078898129)[1];
P2_24 := Points(E24, -14367138002890543477931052222544/12054179879569)[1];
P3_24 := Points(E24, -269944456262034200432888/241081)[1];
P4_24 := Points(E24, -630597721484060507464551524086/528521730025)[1];
P5_24 := Points(E24, 13457909758544525530407577328967629341502974335624109249/9109819423915178769710709651398624256)[1];
E8 := IsogenousCurves(E24)[1]; Coefficients(E8); b, m := IsIsogenous(E24, E8);
P1 := m(P1_24); P2 := m(P2_24); P3 := m(P3_24); P4 := m(P4_24); P5 := m(P5_24);
S := [P1, P2, P3, P4, P5]; T := Points(E8, Saturation(S, 0)[1][1])[1];
P1 := P1+7*T; P2 := P2+3*T; P4 := P4+2*T; P5 := P5+5*T;
P1; P2; P3; P4; P5;
IsLinearlyIndependent(S);
[ 1, 0, 0, -885470498073167713002317184212739304, 315787360224933400897171748517120652503410335938363968 ]
(19736503637532263590768972383399786896/686322068333254991401 : -9688380190239124552471385562273840631362670461480086958584/17980093484565761049363798777899 : 1)
(-106392015003516680180229247909564/262444244417161 : 3315524468078731464225753019750095494243972085244/4251631139754026848091 : 1)
(197824762218747711268509656464/24841227321 : -87398782145351238928127905114416134835854072/3915250679290131 : 1)
(104082169729287819444025464178104869085812/94320856150181594751169 : 23929183871232548960786573517019430438238066845311306934645336/28967539063327699053963692450740447 : 1)
(904856066797346401279199736603573928704209102734767139366416265493676502088160396747547111952/1980903839660916800788793856501687874959261129995118904651263730123078358161 : 7176580364279680654978748955564315110684742734823615940436244933109959252724557389297041035780397662180996315752417825806484247739820234728/88164772399749168946841346769450081472444911489955953664661823265263508750316122576099738571727015984657590089159 : 1)
true
$\endgroup$

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