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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 73689.w
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
73689.w1 | 73689a2 | \([1, 1, 0, -2004, 31365]\) | \(669233048723/50759541\) | \(67560949071\) | \([2]\) | \(55296\) | \(0.82305\) | |
73689.w2 | 73689a1 | \([1, 1, 0, -409, -2768]\) | \(5706550403/1112643\) | \(1480927833\) | \([2]\) | \(27648\) | \(0.47648\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 73689.w have rank \(0\).
Complex multiplication
The elliptic curves in class 73689.w do not have complex multiplication.Modular form 73689.2.a.w
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.