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SageMath
E = EllipticCurve("ez1")
E.isogeny_class()
Elliptic curves in class 194040.ez
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 194040.ez1 | 194040cl3 | \([0, 0, 0, -7244307, 7504873614]\) | \(239369344910082/385\) | \(67624871086080\) | \([2]\) | \(3145728\) | \(2.3470\) | |
| 194040.ez2 | 194040cl4 | \([0, 0, 0, -576387, 48212766]\) | \(120564797922/64054375\) | \(11251087926946560000\) | \([2]\) | \(3145728\) | \(2.3470\) | |
| 194040.ez3 | 194040cl2 | \([0, 0, 0, -452907, 117188694]\) | \(116986321764/148225\) | \(13017787684070400\) | \([2, 2]\) | \(1572864\) | \(2.0004\) | |
| 194040.ez4 | 194040cl1 | \([0, 0, 0, -20727, 2833866]\) | \(-44851536/132055\) | \(-2899416347815680\) | \([2]\) | \(786432\) | \(1.6539\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 194040.ez have rank \(1\).
Complex multiplication
The elliptic curves in class 194040.ez do not have complex multiplication.Modular form 194040.2.a.ez
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.
