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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 162240.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
162240.i1 | 162240dk1 | \([0, -1, 0, -39364381, -84592286819]\) | \(621217777580032/74733890625\) | \(811535868185254992000000\) | \([2]\) | \(23482368\) | \(3.3185\) | \(\Gamma_0(N)\)-optimal |
162240.i2 | 162240dk2 | \([0, -1, 0, 56732399, -433250624015]\) | \(116227003261808/533935546875\) | \(-92768160515004000000000000\) | \([2]\) | \(46964736\) | \(3.6651\) |
Rank
sage: E.rank()
The elliptic curves in class 162240.i have rank \(1\).
Complex multiplication
The elliptic curves in class 162240.i do not have complex multiplication.Modular form 162240.2.a.i
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.