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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 333270.o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
333270.o1 | 333270o3 | \([1, -1, 0, -467606475, 3892072260325]\) | \(104778147797811105409/289854482400\) | \(31280563301402044274400\) | \([2]\) | \(86507520\) | \(3.5492\) | |
333270.o2 | 333270o2 | \([1, -1, 0, -29594475, 59204453125]\) | \(26562019806177409/1343744640000\) | \(145014453198739635840000\) | \([2, 2]\) | \(43253760\) | \(3.2026\) | |
333270.o3 | 333270o1 | \([1, -1, 0, -5218155, -3398811899]\) | \(145606291302529/37984665600\) | \(4099235336798050713600\) | \([2]\) | \(21626880\) | \(2.8560\) | \(\Gamma_0(N)\)-optimal |
333270.o4 | 333270o4 | \([1, -1, 0, 18396405, 232653091621]\) | \(6380108151242111/220374787500000\) | \(-23782442256365720287500000\) | \([2]\) | \(86507520\) | \(3.5492\) |
Rank
sage: E.rank()
The elliptic curves in class 333270.o have rank \(1\).
Complex multiplication
The elliptic curves in class 333270.o do not have complex multiplication.Modular form 333270.2.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 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.