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SageMath
E = EllipticCurve("ew1")
E.isogeny_class()
Elliptic curves in class 22050ew
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
22050.cx2 | 22050ew1 | \([1, -1, 1, -15980, -767833]\) | \(505318200625/4251528\) | \(3796720792200\) | \([]\) | \(69120\) | \(1.2392\) | \(\Gamma_0(N)\)-optimal |
22050.cx1 | 22050ew2 | \([1, -1, 1, -1291730, -564751393]\) | \(266916252066900625/162\) | \(144670050\) | \([]\) | \(207360\) | \(1.7885\) |
Rank
sage: E.rank()
The elliptic curves in class 22050ew have rank \(0\).
Complex multiplication
The elliptic curves in class 22050ew do not have complex multiplication.Modular form 22050.2.a.ew
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.