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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 8550.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8550.e1 | 8550l4 | \([1, -1, 0, -1682442, 811529716]\) | \(46237740924063961/1806561830400\) | \(20577868349400000000\) | \([2]\) | \(165888\) | \(2.4731\) | |
| 8550.e2 | 8550l2 | \([1, -1, 0, -248067, -47154659]\) | \(148212258825961/1218375000\) | \(13878052734375000\) | \([2]\) | \(55296\) | \(1.9238\) | |
| 8550.e3 | 8550l1 | \([1, -1, 0, -5067, -1713659]\) | \(-1263214441/110808000\) | \(-1262172375000000\) | \([2]\) | \(27648\) | \(1.5772\) | \(\Gamma_0(N)\)-optimal |
| 8550.e4 | 8550l3 | \([1, -1, 0, 45558, 46025716]\) | \(918046641959/80912056320\) | \(-921638891520000000\) | \([2]\) | \(82944\) | \(2.1266\) |
Rank
sage: E.rank()
The elliptic curves in class 8550.e have rank \(1\).
Complex multiplication
The elliptic curves in class 8550.e do not have complex multiplication.Modular form 8550.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
