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SageMath
E = EllipticCurve("dk1")
E.isogeny_class()
Elliptic curves in class 28050dk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
28050.cz1 | 28050dk1 | \([1, 0, 0, -2213, -34083]\) | \(76711450249/12622500\) | \(197226562500\) | \([2]\) | \(46080\) | \(0.88889\) | \(\Gamma_0(N)\)-optimal |
28050.cz2 | 28050dk2 | \([1, 0, 0, 4037, -190333]\) | \(465664585751/1274620050\) | \(-19915938281250\) | \([2]\) | \(92160\) | \(1.2355\) |
Rank
sage: E.rank()
The elliptic curves in class 28050dk have rank \(0\).
Complex multiplication
The elliptic curves in class 28050dk do not have complex multiplication.Modular form 28050.2.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.