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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 184338.cp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
184338.cp1 | 184338ew1 | \([1, -1, 0, -9123, -333677]\) | \(-26436959739/50578\) | \(-160662180294\) | \([]\) | \(414720\) | \(1.0397\) | \(\Gamma_0(N)\)-optimal |
184338.cp2 | 184338ew2 | \([1, -1, 0, 15132, -1653688]\) | \(165469149/603592\) | \(-1397729101679064\) | \([]\) | \(1244160\) | \(1.5890\) |
Rank
sage: E.rank()
The elliptic curves in class 184338.cp have rank \(1\).
Complex multiplication
The elliptic curves in class 184338.cp do not have complex multiplication.Modular form 184338.2.a.cp
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 LMFDB 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 LMFDB labels.