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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 5202.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5202.d1 | 5202a4 | \([1, -1, 0, -293967, 42466387]\) | \(159661140625/48275138\) | \(849463221880991538\) | \([2]\) | \(82944\) | \(2.1454\) | |
| 5202.d2 | 5202a3 | \([1, -1, 0, -267957, 53447809]\) | \(120920208625/19652\) | \(345802247865252\) | \([2]\) | \(41472\) | \(1.7988\) | |
| 5202.d3 | 5202a2 | \([1, -1, 0, -111897, -14375867]\) | \(8805624625/2312\) | \(40682617395912\) | \([2]\) | \(27648\) | \(1.5961\) | |
| 5202.d4 | 5202a1 | \([1, -1, 0, -7857, -164003]\) | \(3048625/1088\) | \(19144761127488\) | \([2]\) | \(13824\) | \(1.2495\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 5202.d have rank \(0\).
Complex multiplication
The elliptic curves in class 5202.d do not have complex multiplication.Modular form 5202.2.a.d
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.
