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SageMath
E = EllipticCurve("fa1")
E.isogeny_class()
Elliptic curves in class 64350.fa
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 64350.fa1 | 64350dq4 | \([1, -1, 1, -5266355, 4653009897]\) | \(1418098748958579169/8307406250\) | \(94626549316406250\) | \([2]\) | \(2654208\) | \(2.4470\) | |
| 64350.fa2 | 64350dq3 | \([1, -1, 1, -323105, 75560397]\) | \(-327495950129089/26547449500\) | \(-302392041960937500\) | \([2]\) | \(1327104\) | \(2.1005\) | |
| 64350.fa3 | 64350dq2 | \([1, -1, 1, -93605, 257397]\) | \(7962857630209/4606058600\) | \(52465886240625000\) | \([2]\) | \(884736\) | \(1.8977\) | |
| 64350.fa4 | 64350dq1 | \([1, -1, 1, 23395, 23397]\) | \(124326214271/71980480\) | \(-819902655000000\) | \([2]\) | \(442368\) | \(1.5511\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 64350.fa have rank \(1\).
Complex multiplication
The elliptic curves in class 64350.fa do not have complex multiplication.Modular form 64350.2.a.fa
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.
