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SageMath
E = EllipticCurve("dw1")
E.isogeny_class()
Elliptic curves in class 350658.dw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 350658.dw1 | 350658dw6 | \([1, -1, 1, -86176949, -307893151695]\) | \(54804145548726848737/637608031452\) | \(823450349397402560988\) | \([2]\) | \(41943040\) | \(3.1642\) | |
| 350658.dw2 | 350658dw3 | \([1, -1, 1, -19290569, 32615063385]\) | \(614716917569296417/19093020912\) | \(24658024939295167728\) | \([2]\) | \(20971520\) | \(2.8177\) | |
| 350658.dw3 | 350658dw4 | \([1, -1, 1, -5525609, -4547331687]\) | \(14447092394873377/1439452851984\) | \(1859007251223034100496\) | \([2, 2]\) | \(20971520\) | \(2.8177\) | |
| 350658.dw4 | 350658dw2 | \([1, -1, 1, -1256729, 464333433]\) | \(169967019783457/26337394944\) | \(34013901957078548736\) | \([2, 2]\) | \(10485760\) | \(2.4711\) | |
| 350658.dw5 | 350658dw1 | \([1, -1, 1, 137191, 40024185]\) | \(221115865823/664731648\) | \(-858479631372582912\) | \([2]\) | \(5242880\) | \(2.1245\) | \(\Gamma_0(N)\)-optimal |
| 350658.dw6 | 350658dw5 | \([1, -1, 1, 6823651, -21999305919]\) | \(27207619911317663/177609314617308\) | \(-229376740824296775007452\) | \([2]\) | \(41943040\) | \(3.1642\) |
Rank
sage: E.rank()
The elliptic curves in class 350658.dw have rank \(1\).
Complex multiplication
The elliptic curves in class 350658.dw do not have complex multiplication.Modular form 350658.2.a.dw
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
