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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 182070.bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
182070.bf1 | 182070cw5 | \([1, -1, 0, -43696854, 111190080078]\) | \(524388516989299201/3150\) | \(55428306573150\) | \([2]\) | \(10485760\) | \(2.7027\) | |
182070.bf2 | 182070cw3 | \([1, -1, 0, -2731104, 1737789228]\) | \(128031684631201/9922500\) | \(174599165705422500\) | \([2, 2]\) | \(5242880\) | \(2.3561\) | |
182070.bf3 | 182070cw6 | \([1, -1, 0, -2549034, 1979323290]\) | \(-104094944089921/35880468750\) | \(-631363054559786718750\) | \([2]\) | \(10485760\) | \(2.7027\) | |
182070.bf4 | 182070cw4 | \([1, -1, 0, -962424, -343239660]\) | \(5602762882081/345888060\) | \(6086345850689556060\) | \([2]\) | \(5242880\) | \(2.3561\) | |
182070.bf5 | 182070cw2 | \([1, -1, 0, -182124, 23345280]\) | \(37966934881/8643600\) | \(152095273236723600\) | \([2, 2]\) | \(2621440\) | \(2.0095\) | |
182070.bf6 | 182070cw1 | \([1, -1, 0, 25956, 2245968]\) | \(109902239/188160\) | \(-3310917512636160\) | \([2]\) | \(1310720\) | \(1.6629\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 182070.bf have rank \(1\).
Complex multiplication
The elliptic curves in class 182070.bf do not have complex multiplication.Modular form 182070.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.