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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 34969.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
34969.i1 | 34969d2 | \([0, -1, 1, -3473587, 5701223692]\) | \(-108394872832/265513259\) | \(-11353659165298918896131\) | \([]\) | \(1658880\) | \(2.9200\) | |
34969.i2 | 34969d1 | \([0, -1, 1, 373003, -174442533]\) | \(134217728/384659\) | \(-16448471150982018731\) | \([]\) | \(552960\) | \(2.3707\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 34969.i have rank \(0\).
Complex multiplication
The elliptic curves in class 34969.i do not have complex multiplication.Modular form 34969.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.