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SageMath
E = EllipticCurve("bz1")
E.isogeny_class()
Elliptic curves in class 7650bz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 7650.ca5 | 7650bz1 | \([1, -1, 1, -7655, -237153]\) | \(4354703137/352512\) | \(4015332000000\) | \([2]\) | \(16384\) | \(1.1614\) | \(\Gamma_0(N)\)-optimal |
| 7650.ca4 | 7650bz2 | \([1, -1, 1, -25655, 1310847]\) | \(163936758817/30338064\) | \(345569510250000\) | \([2, 2]\) | \(32768\) | \(1.5080\) | |
| 7650.ca2 | 7650bz3 | \([1, -1, 1, -390155, 93893847]\) | \(576615941610337/27060804\) | \(308239470562500\) | \([2, 2]\) | \(65536\) | \(1.8545\) | |
| 7650.ca6 | 7650bz4 | \([1, -1, 1, 50845, 7583847]\) | \(1276229915423/2927177028\) | \(-33342375834562500\) | \([2]\) | \(65536\) | \(1.8545\) | |
| 7650.ca1 | 7650bz5 | \([1, -1, 1, -6242405, 6004666347]\) | \(2361739090258884097/5202\) | \(59254031250\) | \([2]\) | \(131072\) | \(2.2011\) | |
| 7650.ca3 | 7650bz6 | \([1, -1, 1, -369905, 104059347]\) | \(-491411892194497/125563633938\) | \(-1430248267825031250\) | \([2]\) | \(131072\) | \(2.2011\) |
Rank
sage: E.rank()
The elliptic curves in class 7650bz have rank \(0\).
Complex multiplication
The elliptic curves in class 7650bz do not have complex multiplication.Modular form 7650.2.a.bz
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
