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SageMath
E = EllipticCurve("dk1")
E.isogeny_class()
Elliptic curves in class 321600dk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
321600.dk2 | 321600dk1 | \([0, -1, 0, 23867, -223613]\) | \(1503484706816/890163675\) | \(-890163675000000\) | \([]\) | \(1658880\) | \(1.5579\) | \(\Gamma_0(N)\)-optimal |
321600.dk1 | 321600dk2 | \([0, -1, 0, -300133, 70003387]\) | \(-2989967081734144/380653171875\) | \(-380653171875000000\) | \([]\) | \(4976640\) | \(2.1072\) |
Rank
sage: E.rank()
The elliptic curves in class 321600dk have rank \(0\).
Complex multiplication
The elliptic curves in class 321600dk do not have complex multiplication.Modular form 321600.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.