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SageMath
E = EllipticCurve("bw1")
E.isogeny_class()
Elliptic curves in class 119658bw
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
119658.bu4 | 119658bw1 | \([1, 0, 1, -1116540, -391720262]\) | \(1308451928740468777/194033737531392\) | \(22827875186830737408\) | \([2]\) | \(4423680\) | \(2.4391\) | \(\Gamma_0(N)\)-optimal |
119658.bu2 | 119658bw2 | \([1, 0, 1, -17172860, -27392027974]\) | \(4760617885089919932457/133756441657344\) | \(15736311604544864256\) | \([2, 2]\) | \(8847360\) | \(2.7857\) | |
119658.bu3 | 119658bw3 | \([1, 0, 1, -16482940, -29693601094]\) | \(-4209586785160189454377/801182513521564416\) | \(-94258321533298531977984\) | \([2]\) | \(17694720\) | \(3.1323\) | |
119658.bu1 | 119658bw4 | \([1, 0, 1, -274763900, -1753045923142]\) | \(19499096390516434897995817/15393430272\) | \(1811021678070528\) | \([2]\) | \(17694720\) | \(3.1323\) |
Rank
sage: E.rank()
The elliptic curves in class 119658bw have rank \(1\).
Complex multiplication
The elliptic curves in class 119658bw do not have complex multiplication.Modular form 119658.2.a.bw
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.