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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 5610.bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 5610.bc1 | 5610bb5 | \([1, 1, 1, -292140, -60890853]\) | \(2757381641970898311361/379829992662450\) | \(379829992662450\) | \([2]\) | \(49152\) | \(1.8152\) | |
| 5610.bc2 | 5610bb3 | \([1, 1, 1, -19890, -778053]\) | \(870220733067747361/247623269602500\) | \(247623269602500\) | \([2, 2]\) | \(24576\) | \(1.4686\) | |
| 5610.bc3 | 5610bb2 | \([1, 1, 1, -7390, 231947]\) | \(44633474953947361/1967006250000\) | \(1967006250000\) | \([2, 4]\) | \(12288\) | \(1.1220\) | |
| 5610.bc4 | 5610bb1 | \([1, 1, 1, -7310, 237515]\) | \(43199583152847841/89760000\) | \(89760000\) | \([4]\) | \(6144\) | \(0.77545\) | \(\Gamma_0(N)\)-optimal |
| 5610.bc5 | 5610bb4 | \([1, 1, 1, 3830, 887195]\) | \(6213165856218719/342407226562500\) | \(-342407226562500\) | \([4]\) | \(24576\) | \(1.4686\) | |
| 5610.bc6 | 5610bb6 | \([1, 1, 1, 52360, -5055253]\) | \(15875306080318016639/20322604533582450\) | \(-20322604533582450\) | \([2]\) | \(49152\) | \(1.8152\) |
Rank
sage: E.rank()
The elliptic curves in class 5610.bc have rank \(0\).
Complex multiplication
The elliptic curves in class 5610.bc do not have complex multiplication.Modular form 5610.2.a.bc
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
