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SageMath
E = EllipticCurve("cy1")
E.isogeny_class()
Elliptic curves in class 405042cy
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 405042.cy4 | 405042cy1 | \([1, 0, 0, -2406614, -1081272540]\) | \(32765849647039657/8229948198912\) | \(387185163602178281472\) | \([2]\) | \(18579456\) | \(2.6604\) | \(\Gamma_0(N)\)-optimal |
| 405042.cy2 | 405042cy2 | \([1, 0, 0, -35791894, -82414491676]\) | \(107784459654566688937/10704361149504\) | \(503596100820588393024\) | \([2, 2]\) | \(37158912\) | \(3.0070\) | |
| 405042.cy3 | 405042cy3 | \([1, 0, 0, -33091614, -95373135396]\) | \(-85183593440646799657/34223681512621656\) | \(-1610083247824698426198936\) | \([2]\) | \(74317824\) | \(3.3536\) | |
| 405042.cy1 | 405042cy4 | \([1, 0, 0, -572656654, -5274648331540]\) | \(441453577446719855661097/4354701912\) | \(204870787942424472\) | \([2]\) | \(74317824\) | \(3.3536\) |
Rank
sage: E.rank()
The elliptic curves in class 405042cy have rank \(1\).
Complex multiplication
The elliptic curves in class 405042cy do not have complex multiplication.Modular form 405042.2.a.cy
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 Cremona 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 Cremona labels.
