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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 42042r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
42042.r2 | 42042r1 | \([1, 1, 0, -319, 3025]\) | \(-30664297/18876\) | \(-2220742524\) | \([2]\) | \(23040\) | \(0.49531\) | \(\Gamma_0(N)\)-optimal |
42042.r1 | 42042r2 | \([1, 1, 0, -5709, 163647]\) | \(174958262857/33462\) | \(3936770838\) | \([2]\) | \(46080\) | \(0.84189\) |
Rank
sage: E.rank()
The elliptic curves in class 42042r have rank \(1\).
Complex multiplication
The elliptic curves in class 42042r do not have complex multiplication.Modular form 42042.2.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.