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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 438702x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
438702.x4 | 438702x1 | \([1, 0, 1, -20092, -1996870]\) | \(-37159393753/49741824\) | \(-1200646708985856\) | \([2]\) | \(1769472\) | \(1.5862\) | \(\Gamma_0(N)\)-optimal* |
438702.x3 | 438702x2 | \([1, 0, 1, -390012, -93737030]\) | \(271808161065433/147476736\) | \(3559729891094784\) | \([2, 2]\) | \(3538944\) | \(1.9328\) | \(\Gamma_0(N)\)-optimal* |
438702.x2 | 438702x3 | \([1, 0, 1, -459372, -58113734]\) | \(444142553850073/196663299888\) | \(4746973970814292272\) | \([2]\) | \(7077888\) | \(2.2794\) | \(\Gamma_0(N)\)-optimal* |
438702.x1 | 438702x4 | \([1, 0, 1, -6239372, -5999250886]\) | \(1112891236915770073/327888\) | \(7914419224272\) | \([2]\) | \(7077888\) | \(2.2794\) |
Rank
sage: E.rank()
The elliptic curves in class 438702x have rank \(0\).
Complex multiplication
The elliptic curves in class 438702x do not have complex multiplication.Modular form 438702.2.a.x
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.