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SageMath
E = EllipticCurve("f1")
E.isogeny_class()
Elliptic curves in class 67760.f
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
67760.f1 | 67760be2 | \([0, -1, 0, -1255638336, 17125985729536]\) | \(-249353795628717731809/14000000\) | \(-12292195672064000000\) | \([]\) | \(19160064\) | \(3.5775\) | |
67760.f2 | 67760be1 | \([0, -1, 0, -15359296, 23949777920]\) | \(-456390127585249/17983078400\) | \(-15789394177061971558400\) | \([]\) | \(6386688\) | \(3.0282\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 67760.f have rank \(1\).
Complex multiplication
The elliptic curves in class 67760.f do not have complex multiplication.Modular form 67760.2.a.f
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.