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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 203280.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
203280.i1 | 203280hc4 | \([0, -1, 0, -597296, -177476880]\) | \(12990838708516/144375\) | \(261907578240000\) | \([2]\) | \(1474560\) | \(1.9200\) | |
203280.i2 | 203280hc2 | \([0, -1, 0, -38276, -2615424]\) | \(13674725584/1334025\) | \(605006505734400\) | \([2, 2]\) | \(737280\) | \(1.5735\) | |
203280.i3 | 203280hc1 | \([0, -1, 0, -8631, 266070]\) | \(2508888064/396165\) | \(11229287417040\) | \([2]\) | \(368640\) | \(1.2269\) | \(\Gamma_0(N)\)-optimal |
203280.i4 | 203280hc3 | \([0, -1, 0, 46424, -12643904]\) | \(6099383804/41507235\) | \(-75297381113687040\) | \([2]\) | \(1474560\) | \(1.9200\) |
Rank
sage: E.rank()
The elliptic curves in class 203280.i have rank \(2\).
Complex multiplication
The elliptic curves in class 203280.i do not have complex multiplication.Modular form 203280.2.a.i
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.