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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 78078j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
78078.w2 | 78078j1 | \([1, 1, 0, -68448, 10736640]\) | \(-7347774183121/6119866368\) | \(-29539426063859712\) | \([2]\) | \(1290240\) | \(1.8582\) | \(\Gamma_0(N)\)-optimal |
78078.w1 | 78078j2 | \([1, 1, 0, -1258208, 542559360]\) | \(45637459887836881/13417633152\) | \(64764352456771968\) | \([2]\) | \(2580480\) | \(2.2048\) |
Rank
sage: E.rank()
The elliptic curves in class 78078j have rank \(0\).
Complex multiplication
The elliptic curves in class 78078j do not have complex multiplication.Modular form 78078.2.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 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.