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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 4641.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4641.b1 | 4641b5 | \([1, 1, 1, -40579, -3160960]\) | \(7389727131216686257/6115533215337\) | \(6115533215337\) | \([2]\) | \(12288\) | \(1.3819\) | |
| 4641.b2 | 4641b3 | \([1, 1, 1, -3094, -27214]\) | \(3275619238041697/1605271262049\) | \(1605271262049\) | \([2, 2]\) | \(6144\) | \(1.0353\) | |
| 4641.b3 | 4641b2 | \([1, 1, 1, -1649, 24806]\) | \(495909170514577/6224736609\) | \(6224736609\) | \([2, 4]\) | \(3072\) | \(0.68874\) | |
| 4641.b4 | 4641b1 | \([1, 1, 1, -1644, 24972]\) | \(491411892194497/78897\) | \(78897\) | \([4]\) | \(1536\) | \(0.34217\) | \(\Gamma_0(N)\)-optimal |
| 4641.b5 | 4641b4 | \([1, 1, 1, -284, 66302]\) | \(-2533811507137/1904381781393\) | \(-1904381781393\) | \([4]\) | \(6144\) | \(1.0353\) | |
| 4641.b6 | 4641b6 | \([1, 1, 1, 11271, -193848]\) | \(158346567380527343/108665074944153\) | \(-108665074944153\) | \([2]\) | \(12288\) | \(1.3819\) |
Rank
sage: E.rank()
The elliptic curves in class 4641.b have rank \(1\).
Complex multiplication
The elliptic curves in class 4641.b do not have complex multiplication.Modular form 4641.2.a.b
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.
