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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 51870o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 51870.n4 | 51870o1 | \([1, 1, 0, 838, 1991316]\) | \(64959960000599/1712463779819520\) | \(-1712463779819520\) | \([2]\) | \(368640\) | \(1.6020\) | \(\Gamma_0(N)\)-optimal |
| 51870.n3 | 51870o2 | \([1, 1, 0, -232442, 42255444]\) | \(1388898286900270148521/28874886258254400\) | \(28874886258254400\) | \([2, 2]\) | \(737280\) | \(1.9485\) | |
| 51870.n2 | 51870o3 | \([1, 1, 0, -497042, -72104676]\) | \(13580142956944487962921/5625758146983057720\) | \(5625758146983057720\) | \([2]\) | \(1474560\) | \(2.2951\) | |
| 51870.n1 | 51870o4 | \([1, 1, 0, -3700322, 2738185356]\) | \(5603281597654963338329641/2498954761485000\) | \(2498954761485000\) | \([2]\) | \(1474560\) | \(2.2951\) |
Rank
sage: E.rank()
The elliptic curves in class 51870o have rank \(0\).
Complex multiplication
The elliptic curves in class 51870o do not have complex multiplication.Modular form 51870.2.a.o
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
