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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 100010.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
100010.i1 | 100010h2 | \([1, 1, 1, -1275, 15617]\) | \(229232164503601/20004000200\) | \(20004000200\) | \([2]\) | \(108288\) | \(0.71718\) | |
100010.i2 | 100010h1 | \([1, 1, 1, -275, -1583]\) | \(2300490759601/400040000\) | \(400040000\) | \([2]\) | \(54144\) | \(0.37061\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 100010.i have rank \(0\).
Complex multiplication
The elliptic curves in class 100010.i do not have complex multiplication.Modular form 100010.2.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.