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SageMath
E = EllipticCurve("g1")
E.isogeny_class()
Elliptic curves in class 2450.g
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2450.g1 | 2450p2 | \([1, 1, 0, -6150, 183100]\) | \(-349938025/8\) | \(-588245000\) | \([]\) | \(2160\) | \(0.79616\) | |
2450.g2 | 2450p3 | \([1, 1, 0, -3700, -106000]\) | \(-121945/32\) | \(-1470612500000\) | \([]\) | \(3600\) | \(1.0516\) | |
2450.g3 | 2450p1 | \([1, 1, 0, -25, 575]\) | \(-25/2\) | \(-147061250\) | \([]\) | \(720\) | \(0.24686\) | \(\Gamma_0(N)\)-optimal |
2450.g4 | 2450p4 | \([1, 1, 0, 26925, 782125]\) | \(46969655/32768\) | \(-1505907200000000\) | \([]\) | \(10800\) | \(1.6009\) |
Rank
sage: E.rank()
The elliptic curves in class 2450.g have rank \(1\).
Complex multiplication
The elliptic curves in class 2450.g do not have complex multiplication.Modular form 2450.2.a.g
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.
\(\left(\begin{array}{rrrr} 1 & 15 & 3 & 5 \\ 15 & 1 & 5 & 3 \\ 3 & 5 & 1 & 15 \\ 5 & 3 & 15 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.