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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 47610.x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
47610.x1 | 47610w4 | \([1, -1, 0, -525614499, 4638327636393]\) | \(148809678420065817601/20700\) | \(2233905975776700\) | \([2]\) | \(6488064\) | \(3.2701\) | |
47610.x2 | 47610w6 | \([1, -1, 0, -122976729, -449598702465]\) | \(1905890658841300321/293666194803750\) | \(31691916302207807290353750\) | \([2]\) | \(12976128\) | \(3.6166\) | |
47610.x3 | 47610w3 | \([1, -1, 0, -33707979, 68499268785]\) | \(39248884582600321/3935264062500\) | \(424686468863673576562500\) | \([2, 2]\) | \(6488064\) | \(3.2701\) | |
47610.x4 | 47610w2 | \([1, -1, 0, -32850999, 72479598093]\) | \(36330796409313601/428490000\) | \(46241853698577690000\) | \([2, 2]\) | \(3244032\) | \(2.9235\) | |
47610.x5 | 47610w1 | \([1, -1, 0, -1999719, 1194630525]\) | \(-8194759433281/965779200\) | \(-104225117205837715200\) | \([2]\) | \(1622016\) | \(2.5769\) | \(\Gamma_0(N)\)-optimal |
47610.x6 | 47610w5 | \([1, -1, 0, 41849091, 331845880563]\) | \(75108181893694559/484313964843750\) | \(-52266273440413513183593750\) | \([2]\) | \(12976128\) | \(3.6166\) |
Rank
sage: E.rank()
The elliptic curves in class 47610.x have rank \(0\).
Complex multiplication
The elliptic curves in class 47610.x do not have complex multiplication.Modular form 47610.2.a.x
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 4 & 2 & 4 & 8 \\ 8 & 1 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 2 & 4 & 2 & 1 & 2 & 4 \\ 4 & 8 & 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.