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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 61854.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
61854.j1 | 61854i2 | \([1, 0, 1, -87039, -12429272]\) | \(-15107691357361/5067577806\) | \(-24460230162201054\) | \([]\) | \(648000\) | \(1.8566\) | |
61854.j2 | 61854i1 | \([1, 0, 1, -849, 73348]\) | \(-13997521/474336\) | \(-2289529273824\) | \([]\) | \(129600\) | \(1.0519\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 61854.j have rank \(1\).
Complex multiplication
The elliptic curves in class 61854.j do not have complex multiplication.Modular form 61854.2.a.j
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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.