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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 7650cf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 7650.cn3 | 7650cf1 | \([1, -1, 1, -22730, -1146103]\) | \(114013572049/15667200\) | \(178459200000000\) | \([2]\) | \(36864\) | \(1.4614\) | \(\Gamma_0(N)\)-optimal |
| 7650.cn2 | 7650cf2 | \([1, -1, 1, -94730, 10085897]\) | \(8253429989329/936360000\) | \(10665725625000000\) | \([2, 2]\) | \(73728\) | \(1.8080\) | |
| 7650.cn1 | 7650cf3 | \([1, -1, 1, -1471730, 687569897]\) | \(30949975477232209/478125000\) | \(5446142578125000\) | \([2]\) | \(147456\) | \(2.1546\) | |
| 7650.cn4 | 7650cf4 | \([1, -1, 1, 130270, 50585897]\) | \(21464092074671/109596256200\) | \(-1248369855778125000\) | \([2]\) | \(147456\) | \(2.1546\) |
Rank
sage: E.rank()
The elliptic curves in class 7650cf have rank \(0\).
Complex multiplication
The elliptic curves in class 7650cf do not have complex multiplication.Modular form 7650.2.a.cf
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
