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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 112710.m
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 112710.m1 | 112710q3 | \([1, 1, 0, -48012587, -128069780739]\) | \(507102228823216499929/2648775168000\) | \(63934993383086592000\) | \([2]\) | \(9953280\) | \(2.9965\) | |
| 112710.m2 | 112710q4 | \([1, 1, 0, -47180267, -132722948931]\) | \(-481184224995688814809/36713242449000000\) | \(-886168422826466481000000\) | \([2]\) | \(19906560\) | \(3.3431\) | |
| 112710.m3 | 112710q1 | \([1, 1, 0, -843452, -13491456]\) | \(2749236527524969/1587903192720\) | \(38328122879599297680\) | \([2]\) | \(3317760\) | \(2.4472\) | \(\Gamma_0(N)\)-optimal |
| 112710.m4 | 112710q2 | \([1, 1, 0, 3370168, -103662924]\) | \(175381844946241751/101691694692900\) | \(-2454590297376807560100\) | \([2]\) | \(6635520\) | \(2.7938\) |
Rank
sage: E.rank()
The elliptic curves in class 112710.m have rank \(1\).
Complex multiplication
The elliptic curves in class 112710.m do not have complex multiplication.Modular form 112710.2.a.m
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.
