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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 73689j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
73689.f2 | 73689j1 | \([1, 1, 1, -49552, 3436544]\) | \(5706550403/1112643\) | \(2623553992757313\) | \([2]\) | \(304128\) | \(1.6754\) | \(\Gamma_0(N)\)-optimal |
73689.f1 | 73689j2 | \([1, 1, 1, -242547, -42959454]\) | \(669233048723/50759541\) | \(119688342497169831\) | \([2]\) | \(608256\) | \(2.0220\) |
Rank
sage: E.rank()
The elliptic curves in class 73689j have rank \(0\).
Complex multiplication
The elliptic curves in class 73689j do not have complex multiplication.Modular form 73689.2.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.