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SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 4018.i
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4018.i1 | 4018g2 | \([1, 1, 0, -33664887, -75117062635]\) | \(35864681248144538691049/43574618474283008\) | \(5126510288880921608192\) | \([2]\) | \(783360\) | \(3.0760\) | |
4018.i2 | 4018g1 | \([1, 1, 0, -1552247, -1803905515]\) | \(-3515753329334380009/9905620513718272\) | \(-1165386347818440982528\) | \([2]\) | \(391680\) | \(2.7294\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4018.i have rank \(1\).
Complex multiplication
The elliptic curves in class 4018.i do not have complex multiplication.Modular form 4018.2.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.