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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 38646bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
38646.be1 | 38646bc1 | \([1, -1, 1, -1184, -15357]\) | \(251598106297/412224\) | \(300511296\) | \([2]\) | \(19968\) | \(0.52321\) | \(\Gamma_0(N)\)-optimal |
38646.be2 | 38646bc2 | \([1, -1, 1, -824, -25149]\) | \(-84778086457/331891848\) | \(-241949157192\) | \([2]\) | \(39936\) | \(0.86978\) |
Rank
sage: E.rank()
The elliptic curves in class 38646bc have rank \(0\).
Complex multiplication
The elliptic curves in class 38646bc do not have complex multiplication.Modular form 38646.2.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.