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SageMath
E = EllipticCurve("gl1")
E.isogeny_class()
Elliptic curves in class 43200.gl
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality | CM discriminant |
---|---|---|---|---|---|---|---|---|---|
43200.gl1 | 43200ct4 | \([0, 0, 0, -27000, -1707750]\) | \(-12288000\) | \(-177147000000\) | \([]\) | \(62208\) | \(1.2034\) | \(-27\) | |
43200.gl2 | 43200ct3 | \([0, 0, 0, -3000, 63250]\) | \(-12288000\) | \(-243000000\) | \([]\) | \(20736\) | \(0.65413\) | \(-27\) | |
43200.gl3 | 43200ct2 | \([0, 0, 0, 0, -6750]\) | \(0\) | \(-19683000000\) | \([]\) | \(20736\) | \(0.65413\) | \(-3\) | |
43200.gl4 | 43200ct1 | \([0, 0, 0, 0, 250]\) | \(0\) | \(-27000000\) | \([]\) | \(6912\) | \(0.10483\) | \(\Gamma_0(N)\)-optimal | \(-3\) |
Rank
sage: E.rank()
The elliptic curves in class 43200.gl have rank \(0\).
Complex multiplication
Each elliptic curve in class 43200.gl has complex multiplication by an order in the imaginary quadratic field \(\Q(\sqrt{-3}) \).Modular form 43200.2.a.gl
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.
\(\left(\begin{array}{rrrr} 1 & 27 & 3 & 9 \\ 27 & 1 & 9 & 3 \\ 3 & 9 & 1 & 3 \\ 9 & 3 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.