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SageMath
E = EllipticCurve("hd1")
E.isogeny_class()
Elliptic curves in class 95550.hd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 95550.hd1 | 95550gj4 | \([1, 1, 1, -592313, 175180781]\) | \(12501706118329/2570490\) | \(4725243406406250\) | \([2]\) | \(1179648\) | \(2.0047\) | |
| 95550.hd2 | 95550gj2 | \([1, 1, 1, -41063, 2088281]\) | \(4165509529/1368900\) | \(2516401814062500\) | \([2, 2]\) | \(589824\) | \(1.6581\) | |
| 95550.hd3 | 95550gj1 | \([1, 1, 1, -16563, -802719]\) | \(273359449/9360\) | \(17206166250000\) | \([2]\) | \(294912\) | \(1.3116\) | \(\Gamma_0(N)\)-optimal |
| 95550.hd4 | 95550gj3 | \([1, 1, 1, 118187, 14509781]\) | \(99317171591/106616250\) | \(-195988987441406250\) | \([2]\) | \(1179648\) | \(2.0047\) |
Rank
sage: E.rank()
The elliptic curves in class 95550.hd have rank \(1\).
Complex multiplication
The elliptic curves in class 95550.hd do not have complex multiplication.Modular form 95550.2.a.hd
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.
