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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 54450z
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
54450.f2 | 54450z1 | \([1, -1, 0, 53883, 19873541]\) | \(185193/1936\) | \(-180865305843750000\) | \([2]\) | \(768000\) | \(1.9924\) | \(\Gamma_0(N)\)-optimal |
54450.f1 | 54450z2 | \([1, -1, 0, -853617, 282141041]\) | \(736314327/58564\) | \(5471175501773437500\) | \([2]\) | \(1536000\) | \(2.3390\) |
Rank
sage: E.rank()
The elliptic curves in class 54450z have rank \(1\).
Complex multiplication
The elliptic curves in class 54450z do not have complex multiplication.Modular form 54450.2.a.z
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.