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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 16830.cp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
16830.cp1 | 16830br2 | \([1, -1, 1, -347, 2569]\) | \(170676802323/158950\) | \(4291650\) | \([2]\) | \(6144\) | \(0.19547\) | |
16830.cp2 | 16830br1 | \([1, -1, 1, -17, 61]\) | \(-19034163/41140\) | \(-1110780\) | \([2]\) | \(3072\) | \(-0.15110\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 16830.cp have rank \(0\).
Complex multiplication
The elliptic curves in class 16830.cp do not have complex multiplication.Modular form 16830.2.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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.