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SageMath
E = EllipticCurve("fk1")
E.isogeny_class()
Elliptic curves in class 68400fk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 68400.bt4 | 68400fk1 | \([0, 0, 0, 33605925, -159442757750]\) | \(89962967236397039/287450726400000\) | \(-13411301090918400000000000\) | \([2]\) | \(11059200\) | \(3.5047\) | \(\Gamma_0(N)\)-optimal |
| 68400.bt3 | 68400fk2 | \([0, 0, 0, -316602075, -1868107589750]\) | \(75224183150104868881/11219310000000000\) | \(523448127360000000000000000\) | \([2]\) | \(22118400\) | \(3.8513\) | |
| 68400.bt2 | 68400fk3 | \([0, 0, 0, -11885274075, -498726050597750]\) | \(-3979640234041473454886161/1471455901872240\) | \(-68652246557751229440000000\) | \([2]\) | \(55296000\) | \(4.3094\) | |
| 68400.bt1 | 68400fk4 | \([0, 0, 0, -190164402075, -31918461290189750]\) | \(16300610738133468173382620881/2228489100\) | \(103972387449600000000\) | \([2]\) | \(110592000\) | \(4.6560\) |
Rank
sage: E.rank()
The elliptic curves in class 68400fk have rank \(0\).
Complex multiplication
The elliptic curves in class 68400fk do not have complex multiplication.Modular form 68400.2.a.fk
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
