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SageMath
E = EllipticCurve("dr1")
E.isogeny_class()
Elliptic curves in class 286110dr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
286110.dr1 | 286110dr1 | \([1, -1, 1, -230243, 36031007]\) | \(76711450249/12622500\) | \(222109142768122500\) | \([2]\) | \(4423680\) | \(2.0501\) | \(\Gamma_0(N)\)-optimal |
286110.dr2 | 286110dr2 | \([1, -1, 1, 420007, 202234907]\) | \(465664585751/1274620050\) | \(-22428581236725010050\) | \([2]\) | \(8847360\) | \(2.3967\) |
Rank
sage: E.rank()
The elliptic curves in class 286110dr have rank \(0\).
Complex multiplication
The elliptic curves in class 286110dr do not have complex multiplication.Modular form 286110.2.a.dr
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.