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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 48510.bc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 48510.bc1 | 48510bn6 | \([1, -1, 0, -75448494, -252226953000]\) | \(553808571467029327441/12529687500\) | \(1074622694217187500\) | \([2]\) | \(4718592\) | \(2.9840\) | |
| 48510.bc2 | 48510bn4 | \([1, -1, 0, -5214834, 4574888100]\) | \(182864522286982801/463015182960\) | \(39711016206584498160\) | \([2]\) | \(2359296\) | \(2.6374\) | |
| 48510.bc3 | 48510bn3 | \([1, -1, 0, -4720914, -3930710652]\) | \(135670761487282321/643043610000\) | \(55151356063536810000\) | \([2, 2]\) | \(2359296\) | \(2.6374\) | |
| 48510.bc4 | 48510bn5 | \([1, -1, 0, -2295414, -7966257552]\) | \(-15595206456730321/310672490129100\) | \(-26645174379783696221100\) | \([2]\) | \(4718592\) | \(2.9840\) | |
| 48510.bc5 | 48510bn2 | \([1, -1, 0, -452034, 11173140]\) | \(119102750067601/68309049600\) | \(5858602213388601600\) | \([2, 2]\) | \(1179648\) | \(2.2909\) | |
| 48510.bc6 | 48510bn1 | \([1, -1, 0, 112446, 1351188]\) | \(1833318007919/1070530560\) | \(-91815253543157760\) | \([2]\) | \(589824\) | \(1.9443\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 48510.bc have rank \(0\).
Complex multiplication
The elliptic curves in class 48510.bc do not have complex multiplication.Modular form 48510.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 4 & 8 \\ 8 & 1 & 4 & 8 & 2 & 4 \\ 2 & 4 & 1 & 2 & 2 & 4 \\ 4 & 8 & 2 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
