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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 259182.cr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
259182.cr1 | 259182cr1 | \([1, -1, 0, -115637121, 467719650877]\) | \(132413384610108715177/3460710992707584\) | \(4469397397048037319376896\) | \([2]\) | \(49766400\) | \(3.5115\) | \(\Gamma_0(N)\)-optimal |
259182.cr2 | 259182cr2 | \([1, -1, 0, 20967039, 1505337529405]\) | \(789316843088965463/758459755842421248\) | \(-979526480446047653197005312\) | \([2]\) | \(99532800\) | \(3.8581\) |
Rank
sage: E.rank()
The elliptic curves in class 259182.cr have rank \(1\).
Complex multiplication
The elliptic curves in class 259182.cr do not have complex multiplication.Modular form 259182.2.a.cr
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.