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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 51894e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
51894.n2 | 51894e1 | \([1, -1, 0, -3063, -65333]\) | \(-132651/2\) | \(-47925198774\) | \([]\) | \(59940\) | \(0.85248\) | \(\Gamma_0(N)\)-optimal |
51894.n3 | 51894e2 | \([1, -1, 0, 11352, -329608]\) | \(9261/8\) | \(-139749879624984\) | \([]\) | \(179820\) | \(1.4018\) | |
51894.n1 | 51894e3 | \([1, -1, 0, -118383, 20817197]\) | \(-1167051/512\) | \(-80495930663990784\) | \([]\) | \(539460\) | \(1.9511\) |
Rank
sage: E.rank()
The elliptic curves in class 51894e have rank \(0\).
Complex multiplication
The elliptic curves in class 51894e do not have complex multiplication.Modular form 51894.2.a.e
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.