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SageMath
E = EllipticCurve("dn1")
E.isogeny_class()
Elliptic curves in class 139650.dn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
139650.dn1 | 139650gr1 | \([1, 0, 1, -32903526, -72140372552]\) | \(6248109436056487/50429952000\) | \(31797351000576000000000\) | \([2]\) | \(18579456\) | \(3.1450\) | \(\Gamma_0(N)\)-optimal |
139650.dn2 | 139650gr2 | \([1, 0, 1, -10951526, -166885204552]\) | \(-230380217865127/18948168000000\) | \(-11947295700655875000000000\) | \([2]\) | \(37158912\) | \(3.4915\) |
Rank
sage: E.rank()
The elliptic curves in class 139650.dn have rank \(0\).
Complex multiplication
The elliptic curves in class 139650.dn do not have complex multiplication.Modular form 139650.2.a.dn
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.