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SageMath
sage: E = EllipticCurve("r1")
sage: E.isogeny_class()
Elliptic curves in class 72450.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
72450.r1 | 72450be4 | [1, -1, 0, -678298167, -18483604259] | [2] | 53084160 | |
72450.r2 | 72450be2 | [1, -1, 0, -466614792, -3879452132384] | [2] | 17694720 | |
72450.r3 | 72450be1 | [1, -1, 0, -28602792, -63053576384] | [2] | 8847360 | \(\Gamma_0(N)\)-optimal |
72450.r4 | 72450be3 | [1, -1, 0, 169573833, -2374036259] | [2] | 26542080 |
Rank
sage: E.rank()
The elliptic curves in class 72450.r have rank \(1\).
Complex multiplication
The elliptic curves in class 72450.r do not have complex multiplication.Modular form 72450.2.a.r
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.