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SageMath
E = EllipticCurve("b1")
E.isogeny_class()
Elliptic curves in class 2178.b
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2178.b1 | 2178c3 | \([1, -1, 0, -87687, -9934083]\) | \(57736239625/255552\) | \(330037222413888\) | \([2]\) | \(11520\) | \(1.6382\) | |
| 2178.b2 | 2178c4 | \([1, -1, 0, -44127, -19839627]\) | \(-7357983625/127552392\) | \(-164729828637331848\) | \([2]\) | \(23040\) | \(1.9848\) | |
| 2178.b3 | 2178c1 | \([1, -1, 0, -6012, 170748]\) | \(18609625/1188\) | \(1534263947172\) | \([2]\) | \(3840\) | \(1.0889\) | \(\Gamma_0(N)\)-optimal |
| 2178.b4 | 2178c2 | \([1, -1, 0, 4878, 713070]\) | \(9938375/176418\) | \(-227838196155042\) | \([2]\) | \(7680\) | \(1.4354\) |
Rank
sage: E.rank()
The elliptic curves in class 2178.b have rank \(1\).
Complex multiplication
The elliptic curves in class 2178.b do not have complex multiplication.Modular form 2178.2.a.b
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
