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SageMath
E = EllipticCurve("gq1")
E.isogeny_class()
Elliptic curves in class 78400.gq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 78400.gq1 | 78400gy4 | \([0, 0, 0, -20986700, -37003526000]\) | \(2121328796049/120050\) | \(57850930995200000000\) | \([2]\) | \(3538944\) | \(2.8564\) | |
| 78400.gq2 | 78400gy3 | \([0, 0, 0, -6874700, 6483386000]\) | \(74565301329/5468750\) | \(2635337600000000000000\) | \([2]\) | \(3538944\) | \(2.8564\) | |
| 78400.gq3 | 78400gy2 | \([0, 0, 0, -1386700, -508326000]\) | \(611960049/122500\) | \(59031562240000000000\) | \([2, 2]\) | \(1769472\) | \(2.5098\) | |
| 78400.gq4 | 78400gy1 | \([0, 0, 0, 181300, -47334000]\) | \(1367631/2800\) | \(-1349292851200000000\) | \([2]\) | \(884736\) | \(2.1632\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 78400.gq have rank \(1\).
Complex multiplication
The elliptic curves in class 78400.gq do not have complex multiplication.Modular form 78400.2.a.gq
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.
