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SageMath
E = EllipticCurve("fb1")
E.isogeny_class()
Elliptic curves in class 136710fb
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 136710.bb4 | 136710fb1 | \([1, -1, 0, 612540, -207180464]\) | \(296354077829711/387386634240\) | \(-33224648946010583040\) | \([2]\) | \(4147200\) | \(2.4317\) | \(\Gamma_0(N)\)-optimal |
| 136710.bb3 | 136710fb2 | \([1, -1, 0, -3762180, -2017439600]\) | \(68663623745397169/19216056254400\) | \(1648086605857677182400\) | \([2]\) | \(8294400\) | \(2.7783\) | |
| 136710.bb2 | 136710fb3 | \([1, -1, 0, -17486100, -28308272000]\) | \(-6894246873502147249/47925198774000\) | \(-4110358396999935654000\) | \([2]\) | \(12441600\) | \(2.9810\) | |
| 136710.bb1 | 136710fb4 | \([1, -1, 0, -280242720, -1805646601004]\) | \(28379906689597370652529/1357352437500\) | \(116414853394269937500\) | \([2]\) | \(24883200\) | \(3.3276\) |
Rank
sage: E.rank()
The elliptic curves in class 136710fb have rank \(1\).
Complex multiplication
The elliptic curves in class 136710fb do not have complex multiplication.Modular form 136710.2.a.fb
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
