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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 50.b
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
50.b1 | 50b4 | \([1, 1, 1, -3138, -68969]\) | \(-349938025/8\) | \(-78125000\) | \([]\) | \(30\) | \(0.62793\) | |
50.b2 | 50b3 | \([1, 1, 1, -13, -219]\) | \(-25/2\) | \(-19531250\) | \([]\) | \(10\) | \(0.078619\) | |
50.b3 | 50b1 | \([1, 1, 1, -3, 1]\) | \(-121945/32\) | \(-800\) | \([5]\) | \(2\) | \(-0.72610\) | \(\Gamma_0(N)\)-optimal |
50.b4 | 50b2 | \([1, 1, 1, 22, -9]\) | \(46969655/32768\) | \(-819200\) | \([5]\) | \(6\) | \(-0.17679\) |
Rank
sage: E.rank()
The elliptic curves in class 50.b have rank \(0\).
Complex multiplication
The elliptic curves in class 50.b do not have complex multiplication.Modular form 50.2.a.b
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 15 & 5 \\ 3 & 1 & 5 & 15 \\ 15 & 5 & 1 & 3 \\ 5 & 15 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.