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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 67270u
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 67270.t4 | 67270u1 | \([1, 1, 0, 6181613, -9210893139]\) | \(29434650064089479/58353904000000\) | \(-51789304600720624000000\) | \([2]\) | \(5529600\) | \(3.0429\) | \(\Gamma_0(N)\)-optimal |
| 67270.t3 | 67270u2 | \([1, 1, 0, -46558067, -98309308531]\) | \(12575880055729259641/2575179687500000\) | \(2285481451892679687500000\) | \([2]\) | \(11059200\) | \(3.3895\) | |
| 67270.t2 | 67270u3 | \([1, 1, 0, -58565762, 332032494836]\) | \(-25031389351549772521/39185107281510400\) | \(-34776926952720383239782400\) | \([2]\) | \(16588800\) | \(3.5923\) | |
| 67270.t1 | 67270u4 | \([1, 1, 0, -1160717442, 15211741465844]\) | \(194864658842816448209641/127232527708160000\) | \(112919336683926493736960000\) | \([2]\) | \(33177600\) | \(3.9388\) |
Rank
sage: E.rank()
The elliptic curves in class 67270u have rank \(0\).
Complex multiplication
The elliptic curves in class 67270u do not have complex multiplication.Modular form 67270.2.a.u
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
