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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 10560.bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 10560.bc1 | 10560i4 | \([0, -1, 0, -999985, -384558383]\) | \(6749703004355978704/5671875\) | \(92928000000\) | \([2]\) | \(55296\) | \(1.8401\) | |
| 10560.bc2 | 10560i3 | \([0, -1, 0, -62485, -5995883]\) | \(-26348629355659264/24169921875\) | \(-24750000000000\) | \([2]\) | \(27648\) | \(1.4936\) | |
| 10560.bc3 | 10560i2 | \([0, -1, 0, -12625, -498575]\) | \(13584145739344/1195803675\) | \(19592047411200\) | \([2]\) | \(18432\) | \(1.2908\) | |
| 10560.bc4 | 10560i1 | \([0, -1, 0, 875, -36875]\) | \(72268906496/606436875\) | \(-620991360000\) | \([2]\) | \(9216\) | \(0.94425\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 10560.bc have rank \(0\).
Complex multiplication
The elliptic curves in class 10560.bc do not have complex multiplication.Modular form 10560.2.a.bc
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
