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SageMath
E = EllipticCurve("ek1")
E.isogeny_class()
Elliptic curves in class 304920.ek
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 304920.ek1 | 304920ek4 | \([0, 0, 0, -5375667, -4797251426]\) | \(12990838708516/144375\) | \(190930624536960000\) | \([2]\) | \(5898240\) | \(2.4693\) | |
| 304920.ek2 | 304920ek2 | \([0, 0, 0, -344487, -70960934]\) | \(13674725584/1334025\) | \(441049742680377600\) | \([2, 2]\) | \(2949120\) | \(2.1228\) | |
| 304920.ek3 | 304920ek1 | \([0, 0, 0, -77682, 7106209]\) | \(2508888064/396165\) | \(8186150527022160\) | \([4]\) | \(1474560\) | \(1.7762\) | \(\Gamma_0(N)\)-optimal |
| 304920.ek4 | 304920ek3 | \([0, 0, 0, 417813, -340967594]\) | \(6099383804/41507235\) | \(-54891790831877852160\) | \([2]\) | \(5898240\) | \(2.4693\) |
Rank
sage: E.rank()
The elliptic curves in class 304920.ek have rank \(1\).
Complex multiplication
The elliptic curves in class 304920.ek do not have complex multiplication.Modular form 304920.2.a.ek
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
