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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 49725.f
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 49725.f1 | 49725i6 | \([1, -1, 1, -4539155, 3722855222]\) | \(908031902324522977/161726530797\) | \(1842166264859578125\) | \([2]\) | \(1048576\) | \(2.5092\) | |
| 49725.f2 | 49725i4 | \([1, -1, 1, -312530, 45691472]\) | \(296380748763217/92608836489\) | \(1054872528132515625\) | \([2, 2]\) | \(524288\) | \(2.1626\) | |
| 49725.f3 | 49725i2 | \([1, -1, 1, -122405, -15909028]\) | \(17806161424897/668584449\) | \(7615594739390625\) | \([2, 2]\) | \(262144\) | \(1.8160\) | |
| 49725.f4 | 49725i1 | \([1, -1, 1, -121280, -16226278]\) | \(17319700013617/25857\) | \(294527390625\) | \([2]\) | \(131072\) | \(1.4695\) | \(\Gamma_0(N)\)-optimal |
| 49725.f5 | 49725i3 | \([1, -1, 1, 49720, -57219028]\) | \(1193377118543/124806800313\) | \(-1421627459815265625\) | \([2]\) | \(524288\) | \(2.1626\) | |
| 49725.f6 | 49725i5 | \([1, -1, 1, 872095, 308678222]\) | \(6439735268725823/7345472585373\) | \(-83669523667764328125\) | \([2]\) | \(1048576\) | \(2.5092\) |
Rank
sage: E.rank()
The elliptic curves in class 49725.f have rank \(1\).
Complex multiplication
The elliptic curves in class 49725.f do not have complex multiplication.Modular form 49725.2.a.f
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
