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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 450.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
450.e1 | 450e4 | \([1, -1, 1, -28730, -1867103]\) | \(8527173507/200\) | \(61509375000\) | \([2]\) | \(1152\) | \(1.1822\) | |
450.e2 | 450e3 | \([1, -1, 1, -1730, -31103]\) | \(-1860867/320\) | \(-98415000000\) | \([2]\) | \(576\) | \(0.83566\) | |
450.e3 | 450e2 | \([1, -1, 1, -605, 1647]\) | \(57960603/31250\) | \(13183593750\) | \([2]\) | \(384\) | \(0.63293\) | |
450.e4 | 450e1 | \([1, -1, 1, 145, 147]\) | \(804357/500\) | \(-210937500\) | \([2]\) | \(192\) | \(0.28636\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 450.e have rank \(0\).
Complex multiplication
The elliptic curves in class 450.e do not have complex multiplication.Modular form 450.2.a.e
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.