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SageMath
E = EllipticCurve("h1")
E.isogeny_class()
Elliptic curves in class 1650.h
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1650.h1 | 1650h5 | \([1, 0, 1, -4277126, -3405034852]\) | \(553808571467029327441/12529687500\) | \(195776367187500\) | \([2]\) | \(36864\) | \(2.2665\) | |
| 1650.h2 | 1650h4 | \([1, 0, 1, -295626, 61707148]\) | \(182864522286982801/463015182960\) | \(7234612233750000\) | \([2]\) | \(18432\) | \(1.9199\) | |
| 1650.h3 | 1650h3 | \([1, 0, 1, -267626, -53092852]\) | \(135670761487282321/643043610000\) | \(10047556406250000\) | \([2, 2]\) | \(18432\) | \(1.9199\) | |
| 1650.h4 | 1650h6 | \([1, 0, 1, -130126, -107542852]\) | \(-15595206456730321/310672490129100\) | \(-4854257658267187500\) | \([2]\) | \(36864\) | \(2.2665\) | |
| 1650.h5 | 1650h2 | \([1, 0, 1, -25626, 147148]\) | \(119102750067601/68309049600\) | \(1067328900000000\) | \([2, 2]\) | \(9216\) | \(1.5733\) | |
| 1650.h6 | 1650h1 | \([1, 0, 1, 6374, 19148]\) | \(1833318007919/1070530560\) | \(-16727040000000\) | \([2]\) | \(4608\) | \(1.2268\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 1650.h have rank \(1\).
Complex multiplication
The elliptic curves in class 1650.h do not have complex multiplication.Modular form 1650.2.a.h
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
