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SageMath
E = EllipticCurve("ie1")
E.isogeny_class()
Elliptic curves in class 369600.ie
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 369600.ie1 | 369600ie3 | \([0, -1, 0, -147207233, 687500150337]\) | \(86129359107301290313/9166294368\) | \(37545141731328000000\) | \([2]\) | \(47185920\) | \(3.1837\) | |
| 369600.ie2 | 369600ie2 | \([0, -1, 0, -9223233, 10688630337]\) | \(21184262604460873/216872764416\) | \(888310843047936000000\) | \([2, 2]\) | \(23592960\) | \(2.8372\) | |
| 369600.ie3 | 369600ie4 | \([0, -1, 0, -2311233, 26330486337]\) | \(-333345918055753/72923718045024\) | \(-298695549112418304000000\) | \([2]\) | \(47185920\) | \(3.1837\) | |
| 369600.ie4 | 369600ie1 | \([0, -1, 0, -1031233, -133001663]\) | \(29609739866953/15259926528\) | \(62504659058688000000\) | \([2]\) | \(11796480\) | \(2.4906\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 369600.ie have rank \(0\).
Complex multiplication
The elliptic curves in class 369600.ie do not have complex multiplication.Modular form 369600.2.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.
