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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 66270r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
66270.u2 | 66270r1 | \([1, 1, 1, -17718435, 28703282625]\) | \(-57070627168555729/8686141440\) | \(-93629788959910133760\) | \([2]\) | \(4451328\) | \(2.8446\) | \(\Gamma_0(N)\)-optimal |
66270.u1 | 66270r2 | \([1, 1, 1, -283505315, 1837223528897]\) | \(233786904295505523409/5414400\) | \(58362983477337600\) | \([2]\) | \(8902656\) | \(3.1912\) |
Rank
sage: E.rank()
The elliptic curves in class 66270r have rank \(0\).
Complex multiplication
The elliptic curves in class 66270r do not have complex multiplication.Modular form 66270.2.a.r
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.