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SageMath
E = EllipticCurve("gp1")
E.isogeny_class()
Elliptic curves in class 198450gp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 198450.ba3 | 198450gp1 | \([1, -1, 0, -5742, -174084]\) | \(-140625/8\) | \(-1191196125000\) | \([]\) | \(311040\) | \(1.0741\) | \(\Gamma_0(N)\)-optimal |
| 198450.ba4 | 198450gp2 | \([1, -1, 0, 31008, -333334]\) | \(3375/2\) | \(-1953859444031250\) | \([]\) | \(933120\) | \(1.6234\) | |
| 198450.ba2 | 198450gp3 | \([1, -1, 0, -115992, 30916416]\) | \(-1159088625/2097152\) | \(-312264916992000000\) | \([]\) | \(2177280\) | \(2.0470\) | |
| 198450.ba1 | 198450gp4 | \([1, -1, 0, -11875992, 15755604416]\) | \(-189613868625/128\) | \(-125047004418000000\) | \([]\) | \(6531840\) | \(2.5963\) |
Rank
sage: E.rank()
The elliptic curves in class 198450gp have rank \(0\).
Complex multiplication
The elliptic curves in class 198450gp do not have complex multiplication.Modular form 198450.2.a.gp
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 7 & 21 \\ 3 & 1 & 21 & 7 \\ 7 & 21 & 1 & 3 \\ 21 & 7 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
