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SageMath
E = EllipticCurve("dk1")
E.isogeny_class()
Elliptic curves in class 169050dk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
169050.ff2 | 169050dk1 | \([1, 1, 1, -621713, 187859531]\) | \(14457238157881/49990500\) | \(91895817726562500\) | \([2]\) | \(2654208\) | \(2.1180\) | \(\Gamma_0(N)\)-optimal |
169050.ff1 | 169050dk2 | \([1, 1, 1, -903463, 214031]\) | \(44365623586201/25674468750\) | \(47196493343261718750\) | \([2]\) | \(5308416\) | \(2.4645\) |
Rank
sage: E.rank()
The elliptic curves in class 169050dk have rank \(0\).
Complex multiplication
The elliptic curves in class 169050dk do not have complex multiplication.Modular form 169050.2.a.dk
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 Cremona 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 Cremona labels.