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SageMath
E = EllipticCurve("gk1")
E.isogeny_class()
Elliptic curves in class 47040gk
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 47040.dz3 | 47040gk1 | \([0, 1, 0, -11041, -285601]\) | \(4826809/1680\) | \(51812845486080\) | \([2]\) | \(147456\) | \(1.3331\) | \(\Gamma_0(N)\)-optimal |
| 47040.dz2 | 47040gk2 | \([0, 1, 0, -73761, 7479135]\) | \(1439069689/44100\) | \(1360087194009600\) | \([2, 2]\) | \(294912\) | \(1.6797\) | |
| 47040.dz4 | 47040gk3 | \([0, 1, 0, 20319, 25335519]\) | \(30080231/9003750\) | \(-277684468776960000\) | \([2]\) | \(589824\) | \(2.0263\) | |
| 47040.dz1 | 47040gk4 | \([0, 1, 0, -1171361, 487569375]\) | \(5763259856089/5670\) | \(174868353515520\) | \([2]\) | \(589824\) | \(2.0263\) |
Rank
sage: E.rank()
The elliptic curves in class 47040gk have rank \(0\).
Complex multiplication
The elliptic curves in class 47040gk do not have complex multiplication.Modular form 47040.2.a.gk
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
