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SageMath
E = EllipticCurve("dx1")
E.isogeny_class()
Elliptic curves in class 162450dx
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
162450.bn2 | 162450dx1 | \([1, -1, 0, 43372458, -325221298884]\) | \(129205871/729000\) | \(-50910937236125331890625000\) | \([]\) | \(47278080\) | \(3.6147\) | \(\Gamma_0(N)\)-optimal |
162450.bn1 | 162450dx2 | \([1, -1, 0, -2595627792, -50954441095134]\) | \(-27692833539889/35156250\) | \(-2455195661464377502441406250\) | \([]\) | \(141834240\) | \(4.1640\) |
Rank
sage: E.rank()
The elliptic curves in class 162450dx have rank \(1\).
Complex multiplication
The elliptic curves in class 162450dx do not have complex multiplication.Modular form 162450.2.a.dx
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.