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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 8085.u
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8085.u1 | 8085n3 | \([1, 0, 1, -1014424, 111333191]\) | \(981281029968144361/522287841796875\) | \(61446642299560546875\) | \([2]\) | \(221184\) | \(2.4884\) | |
| 8085.u2 | 8085n2 | \([1, 0, 1, -796129, 273046127]\) | \(474334834335054841/607815140625\) | \(71508843479390625\) | \([2, 2]\) | \(110592\) | \(2.1419\) | |
| 8085.u3 | 8085n1 | \([1, 0, 1, -795884, 273222821]\) | \(473897054735271721/779625\) | \(91722101625\) | \([2]\) | \(55296\) | \(1.7953\) | \(\Gamma_0(N)\)-optimal |
| 8085.u4 | 8085n4 | \([1, 0, 1, -581754, 423451627]\) | \(-185077034913624841/551466161890875\) | \(-64879442480299552875\) | \([2]\) | \(221184\) | \(2.4884\) |
Rank
sage: E.rank()
The elliptic curves in class 8085.u have rank \(1\).
Complex multiplication
The elliptic curves in class 8085.u do not have complex multiplication.Modular form 8085.2.a.u
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.
