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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 83370.x
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
83370.x1 | 83370x3 | \([1, 0, 0, -47924887245, -4039213143491223]\) | \(-12173263182155064921392630443321792081/3458470272310068079827810615960\) | \(-3458470272310068079827810615960\) | \([]\) | \(354818880\) | \(4.8406\) | |
83370.x2 | 83370x1 | \([1, 0, 0, -32007045, 750143318337]\) | \(-3626272648660625410108067281/240996070457081856000000000\) | \(-240996070457081856000000000\) | \([9]\) | \(39424320\) | \(3.7420\) | \(\Gamma_0(N)\)-optimal |
83370.x3 | 83370x2 | \([1, 0, 0, 287672955, -20100659817663]\) | \(2632817860539621521875452252719/176067816626824667205829056000\) | \(-176067816626824667205829056000\) | \([3]\) | \(118272960\) | \(4.2913\) |
Rank
sage: E.rank()
The elliptic curves in class 83370.x have rank \(0\).
Complex multiplication
The elliptic curves in class 83370.x do not have complex multiplication.Modular form 83370.2.a.x
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.
\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.