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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 550i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
550.i1 | 550i1 | \([1, 1, 1, -2213, 39531]\) | \(-76711450249/851840\) | \(-13310000000\) | \([]\) | \(672\) | \(0.75753\) | \(\Gamma_0(N)\)-optimal |
550.i2 | 550i2 | \([1, 1, 1, 7412, 212781]\) | \(2882081488391/2883584000\) | \(-45056000000000\) | \([]\) | \(2016\) | \(1.3068\) |
Rank
sage: E.rank()
The elliptic curves in class 550i have rank \(1\).
Complex multiplication
The elliptic curves in class 550i do not have complex multiplication.Modular form 550.2.a.i
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.