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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 4050.d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4050.d1 | 4050e2 | \([1, -1, 0, -582, -5284]\) | \(-1642545/8\) | \(-106288200\) | \([]\) | \(1944\) | \(0.38866\) | |
4050.d2 | 4050e1 | \([1, -1, 0, 18, -44]\) | \(308655/512\) | \(-1036800\) | \([]\) | \(648\) | \(-0.16064\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 4050.d have rank \(1\).
Complex multiplication
The elliptic curves in class 4050.d do not have complex multiplication.Modular form 4050.2.a.d
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.