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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 1089.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
1089.j1 | 1089g3 | \([1, -1, 0, -159561, 24548134]\) | \(347873904937/395307\) | \(510526328421483\) | \([2]\) | \(5760\) | \(1.7356\) | |
1089.j2 | 1089g2 | \([1, -1, 0, -12546, 173047]\) | \(169112377/88209\) | \(113919098077521\) | \([2, 2]\) | \(2880\) | \(1.3890\) | |
1089.j3 | 1089g1 | \([1, -1, 0, -7101, -226616]\) | \(30664297/297\) | \(383565986793\) | \([2]\) | \(1440\) | \(1.0425\) | \(\Gamma_0(N)\)-optimal |
1089.j4 | 1089g4 | \([1, -1, 0, 47349, 1311052]\) | \(9090072503/5845851\) | \(-7549729318046619\) | \([2]\) | \(5760\) | \(1.7356\) |
Rank
sage: E.rank()
The elliptic curves in class 1089.j have rank \(1\).
Complex multiplication
The elliptic curves in class 1089.j do not have complex multiplication.Modular form 1089.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.