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SageMath
E = EllipticCurve("ei1")
E.isogeny_class()
Elliptic curves in class 54720.ei
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 54720.ei1 | 54720el4 | \([0, 0, 0, -30426304332, -2042781522572144]\) | \(16300610738133468173382620881/2228489100\) | \(425870898993561600\) | \([2]\) | \(36864000\) | \(4.1979\) | |
| 54720.ei2 | 54720el3 | \([0, 0, 0, -1901643852, -31918467238256]\) | \(-3979640234041473454886161/1471455901872240\) | \(-281199601900549035786240\) | \([2]\) | \(18432000\) | \(3.8513\) | |
| 54720.ei3 | 54720el2 | \([0, 0, 0, -50656332, -119558885744]\) | \(75224183150104868881/11219310000000000\) | \(2144043529666560000000000\) | \([2]\) | \(7372800\) | \(3.3931\) | |
| 54720.ei4 | 54720el1 | \([0, 0, 0, 5376948, -10204336496]\) | \(89962967236397039/287450726400000\) | \(-54932689268401766400000\) | \([2]\) | \(3686400\) | \(3.0466\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 54720.ei have rank \(0\).
Complex multiplication
The elliptic curves in class 54720.ei do not have complex multiplication.Modular form 54720.2.a.ei
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
