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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 234416.bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 234416.bf1 | 234416bf3 | \([0, 0, 0, -2187899, 1245629882]\) | \(9614292367656708/2093\) | \(252149101568\) | \([2]\) | \(1572864\) | \(2.0121\) | |
| 234416.bf2 | 234416bf4 | \([0, 0, 0, -159299, 12610738]\) | \(3710860803108/1577224103\) | \(190012250617699328\) | \([2]\) | \(1572864\) | \(2.0121\) | |
| 234416.bf3 | 234416bf2 | \([0, 0, 0, -136759, 19458390]\) | \(9392111857872/4380649\) | \(131937017395456\) | \([2, 2]\) | \(786432\) | \(1.6656\) | |
| 234416.bf4 | 234416bf1 | \([0, 0, 0, -7154, 406455]\) | \(-21511084032/25465531\) | \(-47935908105904\) | \([2]\) | \(393216\) | \(1.3190\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 234416.bf have rank \(0\).
Complex multiplication
The elliptic curves in class 234416.bf do not have complex multiplication.Modular form 234416.2.a.bf
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
