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SageMath
E = EllipticCurve("cw1")
E.isogeny_class()
Elliptic curves in class 180336cw
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
180336.v1 | 180336cw1 | \([0, -1, 0, -4129328, -3217155696]\) | \(315042014258500/1262881737\) | \(31214484547253609472\) | \([2]\) | \(4423680\) | \(2.5972\) | \(\Gamma_0(N)\)-optimal |
180336.v2 | 180336cw2 | \([0, -1, 0, -2175688, -6271085744]\) | \(-23040414103250/330419182041\) | \(-16333856369537430177792\) | \([2]\) | \(8847360\) | \(2.9438\) |
Rank
sage: E.rank()
The elliptic curves in class 180336cw have rank \(1\).
Complex multiplication
The elliptic curves in class 180336cw do not have complex multiplication.Modular form 180336.2.a.cw
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.