Show commands:
SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 6930.x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 6930.x1 | 6930bc4 | \([1, -1, 1, -188183, 31467871]\) | \(1010962818911303721/57392720\) | \(41839292880\) | \([2]\) | \(32768\) | \(1.5047\) | |
| 6930.x2 | 6930bc3 | \([1, -1, 1, -19703, -246113]\) | \(1160306142246441/634128110000\) | \(462279392190000\) | \([2]\) | \(32768\) | \(1.5047\) | |
| 6930.x3 | 6930bc2 | \([1, -1, 1, -11783, 492031]\) | \(248158561089321/1859334400\) | \(1355454777600\) | \([2, 2]\) | \(16384\) | \(1.1581\) | |
| 6930.x4 | 6930bc1 | \([1, -1, 1, -263, 17407]\) | \(-2749884201/176619520\) | \(-128755630080\) | \([2]\) | \(8192\) | \(0.81155\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 6930.x have rank \(1\).
Complex multiplication
The elliptic curves in class 6930.x do not have complex multiplication.Modular form 6930.2.a.x
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.
