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SageMath
E = EllipticCurve("fw1")
E.isogeny_class()
Elliptic curves in class 163170fw
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
163170.o2 | 163170fw1 | \([1, -1, 0, -16620, 1002896]\) | \(-219256227/59200\) | \(-137088567806400\) | \([2]\) | \(663552\) | \(1.4289\) | \(\Gamma_0(N)\)-optimal |
163170.o1 | 163170fw2 | \([1, -1, 0, -281220, 57468536]\) | \(1062144635427/54760\) | \(126806925220920\) | \([2]\) | \(1327104\) | \(1.7755\) |
Rank
sage: E.rank()
The elliptic curves in class 163170fw have rank \(1\).
Complex multiplication
The elliptic curves in class 163170fw do not have complex multiplication.Modular form 163170.2.a.fw
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.