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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 139638.bk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 139638.bk1 | 139638g3 | \([1, 0, 0, -1027463, -303153015]\) | \(46753267515625/11591221248\) | \(29739902468555538432\) | \([2]\) | \(3576960\) | \(2.4469\) | |
| 139638.bk2 | 139638g1 | \([1, 0, 0, -349808, 79574208]\) | \(1845026709625/793152\) | \(2035011032751168\) | \([2]\) | \(1192320\) | \(1.8976\) | \(\Gamma_0(N)\)-optimal |
| 139638.bk3 | 139638g2 | \([1, 0, 0, -295048, 105344264]\) | \(-1107111813625/1228691592\) | \(-3152486466110653128\) | \([2]\) | \(2384640\) | \(2.2442\) | |
| 139638.bk4 | 139638g4 | \([1, 0, 0, 2477177, -1921595767]\) | \(655215969476375/1001033261568\) | \(-2568377475492422349312\) | \([2]\) | \(7153920\) | \(2.7935\) |
Rank
sage: E.rank()
The elliptic curves in class 139638.bk have rank \(0\).
Complex multiplication
The elliptic curves in class 139638.bk do not have complex multiplication.Modular form 139638.2.a.bk
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 LMFDB numbering.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
