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SageMath
E = EllipticCurve("gc1")
E.isogeny_class()
Elliptic curves in class 129360.gc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 129360.gc1 | 129360hn4 | \([0, 1, 0, -12249820, -16506315400]\) | \(6749703004355978704/5671875\) | \(170826348000000\) | \([2]\) | \(2488320\) | \(2.4665\) | |
| 129360.gc2 | 129360hn3 | \([0, 1, 0, -765445, -258221650]\) | \(-26348629355659264/24169921875\) | \(-45497074218750000\) | \([2]\) | \(1244160\) | \(2.1199\) | |
| 129360.gc3 | 129360hn2 | \([0, 1, 0, -154660, -21608392]\) | \(13584145739344/1195803675\) | \(36015387279379200\) | \([2]\) | \(829440\) | \(1.9172\) | |
| 129360.gc4 | 129360hn1 | \([0, 1, 0, 10715, -1564942]\) | \(72268906496/606436875\) | \(-1141547070510000\) | \([2]\) | \(414720\) | \(1.5706\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 129360.gc have rank \(1\).
Complex multiplication
The elliptic curves in class 129360.gc do not have complex multiplication.Modular form 129360.2.a.gc
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.
