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SageMath
E = EllipticCurve("bi1")
E.isogeny_class()
Elliptic curves in class 6050.bi
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
6050.bi1 | 6050bi2 | \([1, 0, 0, -15188, 719192]\) | \(-349938025/8\) | \(-8857805000\) | \([]\) | \(8100\) | \(1.0222\) | |
6050.bi2 | 6050bi3 | \([1, 0, 0, -9138, -406108]\) | \(-121945/32\) | \(-22144512500000\) | \([]\) | \(13500\) | \(1.2776\) | |
6050.bi3 | 6050bi1 | \([1, 0, 0, -63, 2267]\) | \(-25/2\) | \(-2214451250\) | \([]\) | \(2700\) | \(0.47285\) | \(\Gamma_0(N)\)-optimal |
6050.bi4 | 6050bi4 | \([1, 0, 0, 66487, 2997017]\) | \(46969655/32768\) | \(-22675980800000000\) | \([]\) | \(40500\) | \(1.8269\) |
Rank
sage: E.rank()
The elliptic curves in class 6050.bi have rank \(0\).
Complex multiplication
The elliptic curves in class 6050.bi do not have complex multiplication.Modular form 6050.2.a.bi
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.