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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 1650.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1650.e1 | 1650g3 | \([1, 0, 1, -45822651, -119386039802]\) | \(680995599504466943307169/52207031250000000\) | \(815734863281250000000\) | \([2]\) | \(215040\) | \(3.0608\) | |
1650.e2 | 1650g2 | \([1, 0, 1, -3054651, -1602967802]\) | \(201738262891771037089/45727545600000000\) | \(714492900000000000000\) | \([2, 2]\) | \(107520\) | \(2.7142\) | |
1650.e3 | 1650g1 | \([1, 0, 1, -1006651, 367208198]\) | \(7220044159551112609/448454983680000\) | \(7007109120000000000\) | \([2]\) | \(53760\) | \(2.3676\) | \(\Gamma_0(N)\)-optimal |
1650.e4 | 1650g4 | \([1, 0, 1, 6945349, -9902967802]\) | \(2371297246710590562911/4084000833203280000\) | \(-63812513018801250000000\) | \([2]\) | \(215040\) | \(3.0608\) |
Rank
sage: E.rank()
The elliptic curves in class 1650.e have rank \(0\).
Complex multiplication
The elliptic curves in class 1650.e do not have complex multiplication.Modular form 1650.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.