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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 42050f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
42050.m3 | 42050f1 | \([1, 0, 1, -2541, 59208]\) | \(-121945/32\) | \(-475858656800\) | \([]\) | \(50400\) | \(0.95755\) | \(\Gamma_0(N)\)-optimal |
42050.m4 | 42050f2 | \([1, 0, 1, 18484, -436982]\) | \(46969655/32768\) | \(-487279264563200\) | \([]\) | \(151200\) | \(1.5069\) | |
42050.m2 | 42050f3 | \([1, 0, 1, -10951, -5205452]\) | \(-25/2\) | \(-11617642988281250\) | \([]\) | \(252000\) | \(1.7623\) | |
42050.m1 | 42050f4 | \([1, 0, 1, -2639076, -1650411702]\) | \(-349938025/8\) | \(-46470571953125000\) | \([]\) | \(756000\) | \(2.3116\) |
Rank
sage: E.rank()
The elliptic curves in class 42050f have rank \(1\).
Complex multiplication
The elliptic curves in class 42050f do not have complex multiplication.Modular form 42050.2.a.f
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 5 & 15 \\ 3 & 1 & 15 & 5 \\ 5 & 15 & 1 & 3 \\ 15 & 5 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.