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SageMath
E = EllipticCurve("ie1")
E.isogeny_class()
Elliptic curves in class 177870.ie
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 177870.ie1 | 177870j3 | \([1, 0, 0, -2214605, 1268320395]\) | \(5763259856089/5670\) | \(1181754895104630\) | \([2]\) | \(3932160\) | \(2.1855\) | |
| 177870.ie2 | 177870j2 | \([1, 0, 0, -139455, 19495125]\) | \(1439069689/44100\) | \(9191426961924900\) | \([2, 2]\) | \(1966080\) | \(1.8389\) | |
| 177870.ie3 | 177870j1 | \([1, 0, 0, -20875, -734623]\) | \(4826809/1680\) | \(350149598549520\) | \([2]\) | \(983040\) | \(1.4923\) | \(\Gamma_0(N)\)-optimal |
| 177870.ie4 | 177870j4 | \([1, 0, 0, 38415, 65848047]\) | \(30080231/9003750\) | \(-1876583004726333750\) | \([2]\) | \(3932160\) | \(2.1855\) |
Rank
sage: E.rank()
The elliptic curves in class 177870.ie have rank \(0\).
Complex multiplication
The elliptic curves in class 177870.ie do not have complex multiplication.Modular form 177870.2.a.ie
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
