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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 429429o
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
429429.o2 | 429429o1 | \([0, 1, 1, 974736, -4223177656]\) | \(5451776/413343\) | \(-7765275508678217308779\) | \([]\) | \(44928000\) | \(2.8799\) | \(\Gamma_0(N)\)-optimal |
429429.o1 | 429429o2 | \([0, 1, 1, -1314918414, -18352984625530]\) | \(-13383627864961024/151263\) | \(-2841704998679530522539\) | \([]\) | \(224640000\) | \(3.6846\) |
Rank
sage: E.rank()
The elliptic curves in class 429429o have rank \(0\).
Complex multiplication
The elliptic curves in class 429429o do not have complex multiplication.Modular form 429429.2.a.o
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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.