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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 4998j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
4998.a2 | 4998j1 | \([1, 1, 0, -266109, -52946739]\) | \(42531320912955257257/1127938881456\) | \(55269005191344\) | \([]\) | \(60480\) | \(1.7422\) | \(\Gamma_0(N)\)-optimal |
4998.a1 | 4998j2 | \([1, 1, 0, -461724, 34374096]\) | \(222165413800219579417/118033833938006016\) | \(5783657862962294784\) | \([]\) | \(181440\) | \(2.2915\) |
Rank
sage: E.rank()
The elliptic curves in class 4998j have rank \(1\).
Complex multiplication
The elliptic curves in class 4998j do not have complex multiplication.Modular form 4998.2.a.j
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.