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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 60690bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
60690.bc2 | 60690bc1 | \([1, 0, 1, -2143678748, -38373422912422]\) | \(-9186763300983704416553/47730830553907200\) | \(-5660297838825981134399078400\) | \([2]\) | \(75202560\) | \(4.1711\) | \(\Gamma_0(N)\)-optimal |
60690.bc1 | 60690bc2 | \([1, 0, 1, -34341515548, -2449501992113062]\) | \(37769548376817211811066153/1011738331054080\) | \(119979900250322138867957760\) | \([2]\) | \(150405120\) | \(4.5176\) |
Rank
sage: E.rank()
The elliptic curves in class 60690bc have rank \(1\).
Complex multiplication
The elliptic curves in class 60690bc do not have complex multiplication.Modular form 60690.2.a.bc
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.