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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 476850.bk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
476850.bk1 | 476850bk6 | \([1, 1, 0, -1236089275, -16727700137375]\) | \(553808571467029327441/12529687500\) | \(4725565571557617187500\) | \([2]\) | \(188743680\) | \(3.6831\) | |
476850.bk2 | 476850bk3 | \([1, 1, 0, -85435775, 303252655125]\) | \(182864522286982801/463015182960\) | \(174625951980384753750000\) | \([2]\) | \(94371840\) | \(3.3365\) | \(\Gamma_0(N)\)-optimal* |
476850.bk3 | 476850bk4 | \([1, 1, 0, -77343775, -260767836875]\) | \(135670761487282321/643043610000\) | \(242523586037251406250000\) | \([2, 2]\) | \(94371840\) | \(3.3365\) | |
476850.bk4 | 476850bk5 | \([1, 1, 0, -37606275, -528320424375]\) | \(-15595206456730321/310672490129100\) | \(-117169979170202658717187500\) | \([2]\) | \(188743680\) | \(3.6831\) | |
476850.bk5 | 476850bk2 | \([1, 1, 0, -7405775, 730345125]\) | \(119102750067601/68309049600\) | \(25762724969444100000000\) | \([2, 2]\) | \(47185920\) | \(2.9899\) | \(\Gamma_0(N)\)-optimal* |
476850.bk6 | 476850bk1 | \([1, 1, 0, 1842225, 92233125]\) | \(1833318007919/1070530560\) | \(-403750082165760000000\) | \([2]\) | \(23592960\) | \(2.6434\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 476850.bk have rank \(1\).
Complex multiplication
The elliptic curves in class 476850.bk do not have complex multiplication.Modular form 476850.2.a.bk
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.