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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 250470.r
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 250470.r1 | 250470r4 | \([1, -1, 0, -1603575, -781194375]\) | \(353108405631241/172500\) | \(222778224652500\) | \([2]\) | \(2621440\) | \(2.0882\) | |
| 250470.r2 | 250470r2 | \([1, -1, 0, -100755, -12051099]\) | \(87587538121/1904400\) | \(2459471600163600\) | \([2, 2]\) | \(1310720\) | \(1.7416\) | |
| 250470.r3 | 250470r1 | \([1, -1, 0, -13635, 337365]\) | \(217081801/88320\) | \(114062451022080\) | \([2]\) | \(655360\) | \(1.3950\) | \(\Gamma_0(N)\)-optimal |
| 250470.r4 | 250470r3 | \([1, -1, 0, 8145, -36814959]\) | \(46268279/453342420\) | \(-585477214418944980\) | \([2]\) | \(2621440\) | \(2.0882\) |
Rank
sage: E.rank()
The elliptic curves in class 250470.r have rank \(1\).
Complex multiplication
The elliptic curves in class 250470.r do not have complex multiplication.Modular form 250470.2.a.r
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.
