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SageMath
E = EllipticCurve("iw1")
E.isogeny_class()
Elliptic curves in class 177450iw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 177450.bw3 | 177450iw1 | \([1, 1, 0, -11300, -466500]\) | \(4649101309/6804\) | \(233568562500\) | \([2]\) | \(307200\) | \(1.0830\) | \(\Gamma_0(N)\)-optimal |
| 177450.bw4 | 177450iw2 | \([1, 1, 0, -8050, -736250]\) | \(-1680914269/5786802\) | \(-198650062406250\) | \([2]\) | \(614400\) | \(1.4296\) | |
| 177450.bw1 | 177450iw3 | \([1, 1, 0, -337925, 75442125]\) | \(124318741396429/51631104\) | \(1772398992000000\) | \([2]\) | \(1536000\) | \(1.8878\) | |
| 177450.bw2 | 177450iw4 | \([1, 1, 0, -285925, 99518125]\) | \(-75306487574989/81352871712\) | \(-2792691549238500000\) | \([2]\) | \(3072000\) | \(2.2343\) |
Rank
sage: E.rank()
The elliptic curves in class 177450iw have rank \(1\).
Complex multiplication
The elliptic curves in class 177450iw do not have complex multiplication.Modular form 177450.2.a.iw
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 Cremona 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 Cremona labels.
