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SageMath
E = EllipticCurve("u1")
E.isogeny_class()
Elliptic curves in class 121242u
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
121242.y2 | 121242u1 | \([1, 1, 1, -2107883, 1177841369]\) | \(-584665029122631625/456304451328\) | \(-808371170099083008\) | \([2]\) | \(2764800\) | \(2.3674\) | \(\Gamma_0(N)\)-optimal |
121242.y1 | 121242u2 | \([1, 1, 1, -33732443, 75394358777]\) | \(2396133370390615719625/1457813808\) | \(2582606087514288\) | \([2]\) | \(5529600\) | \(2.7140\) |
Rank
sage: E.rank()
The elliptic curves in class 121242u have rank \(1\).
Complex multiplication
The elliptic curves in class 121242u do not have complex multiplication.Modular form 121242.2.a.u
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.