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SageMath
E = EllipticCurve("ko1")
E.isogeny_class()
Elliptic curves in class 176400ko
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
176400.pt2 | 176400ko1 | \([0, 0, 0, -222075, -38697750]\) | \(2803221/125\) | \(54010152000000000\) | \([2]\) | \(1769472\) | \(1.9733\) | \(\Gamma_0(N)\)-optimal |
176400.pt1 | 176400ko2 | \([0, 0, 0, -600075, 128000250]\) | \(55306341/15625\) | \(6751269000000000000\) | \([2]\) | \(3538944\) | \(2.3199\) |
Rank
sage: E.rank()
The elliptic curves in class 176400ko have rank \(1\).
Complex multiplication
The elliptic curves in class 176400ko do not have complex multiplication.Modular form 176400.2.a.ko
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.