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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 62475.cm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 62475.cm1 | 62475bs4 | \([1, 0, 1, -2065376, -1142597977]\) | \(530044731605089/26309115\) | \(48363141728671875\) | \([2]\) | \(1179648\) | \(2.2728\) | |
| 62475.cm2 | 62475bs3 | \([1, 0, 1, -656626, 190324523]\) | \(17032120495489/1339001685\) | \(2461440769352578125\) | \([2]\) | \(1179648\) | \(2.2728\) | |
| 62475.cm3 | 62475bs2 | \([1, 0, 1, -136001, -15842977]\) | \(151334226289/28676025\) | \(52714151019140625\) | \([2, 2]\) | \(589824\) | \(1.9263\) | |
| 62475.cm4 | 62475bs1 | \([1, 0, 1, 17124, -1449227]\) | \(302111711/669375\) | \(-1230489052734375\) | \([2]\) | \(294912\) | \(1.5797\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 62475.cm have rank \(1\).
Complex multiplication
The elliptic curves in class 62475.cm do not have complex multiplication.Modular form 62475.2.a.cm
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
