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SageMath
E = EllipticCurve("gv1")
E.isogeny_class()
Elliptic curves in class 193200.gv
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 193200.gv1 | 193200bk4 | \([0, 1, 0, -1205863408, -44616896812]\) | \(3029968325354577848895529/1753440696000000000000\) | \(112220204544000000000000000000\) | \([2]\) | \(159252480\) | \(4.2637\) | |
| 193200.gv2 | 193200bk2 | \([0, 1, 0, -829537408, -9196291412812]\) | \(986396822567235411402169/6336721794060000\) | \(405550194819840000000000\) | \([2]\) | \(53084160\) | \(3.7143\) | |
| 193200.gv3 | 193200bk1 | \([0, 1, 0, -50849408, -149494228812]\) | \(-227196402372228188089/19338934824115200\) | \(-1237691828743372800000000\) | \([2]\) | \(26542080\) | \(3.3678\) | \(\Gamma_0(N)\)-optimal |
| 193200.gv4 | 193200bk3 | \([0, 1, 0, 301464592, -5426368812]\) | \(47342661265381757089751/27397579603968000000\) | \(-1753445094653952000000000000\) | \([2]\) | \(79626240\) | \(3.9171\) |
Rank
sage: E.rank()
The elliptic curves in class 193200.gv have rank \(1\).
Complex multiplication
The elliptic curves in class 193200.gv do not have complex multiplication.Modular form 193200.2.a.gv
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
