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SageMath
E = EllipticCurve("fp1")
E.isogeny_class()
Elliptic curves in class 22050fp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
22050.dm4 | 22050fp1 | \([1, -1, 1, -33305, -3173803]\) | \(-24389/12\) | \(-2010143460937500\) | \([2]\) | \(115200\) | \(1.6417\) | \(\Gamma_0(N)\)-optimal |
22050.dm2 | 22050fp2 | \([1, -1, 1, -584555, -171856303]\) | \(131872229/18\) | \(3015215191406250\) | \([2]\) | \(230400\) | \(1.9883\) | |
22050.dm3 | 22050fp3 | \([1, -1, 1, -308930, 317653697]\) | \(-19465109/248832\) | \(-41682334806000000000\) | \([2]\) | \(576000\) | \(2.4464\) | |
22050.dm1 | 22050fp4 | \([1, -1, 1, -9128930, 10584133697]\) | \(502270291349/1889568\) | \(316525229933062500000\) | \([2]\) | \(1152000\) | \(2.7930\) |
Rank
sage: E.rank()
The elliptic curves in class 22050fp have rank \(1\).
Complex multiplication
The elliptic curves in class 22050fp do not have complex multiplication.Modular form 22050.2.a.fp
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.