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SageMath
E = EllipticCurve("dh1")
E.isogeny_class()
Elliptic curves in class 22050dh
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 22050.ft4 | 22050dh1 | \([1, -1, 1, 7120, -64753]\) | \(804357/500\) | \(-24816585937500\) | \([2]\) | \(69120\) | \(1.2593\) | \(\Gamma_0(N)\)-optimal |
| 22050.ft3 | 22050dh2 | \([1, -1, 1, -29630, -505753]\) | \(57960603/31250\) | \(1551036621093750\) | \([2]\) | \(138240\) | \(1.6059\) | |
| 22050.ft2 | 22050dh3 | \([1, -1, 1, -84755, 10837747]\) | \(-1860867/320\) | \(-11578426335000000\) | \([2]\) | \(207360\) | \(1.8086\) | |
| 22050.ft1 | 22050dh4 | \([1, -1, 1, -1407755, 643231747]\) | \(8527173507/200\) | \(7236516459375000\) | \([2]\) | \(414720\) | \(2.1552\) |
Rank
sage: E.rank()
The elliptic curves in class 22050dh have rank \(1\).
Complex multiplication
The elliptic curves in class 22050dh do not have complex multiplication.Modular form 22050.2.a.dh
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
