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SageMath
E = EllipticCurve("ck1")
E.isogeny_class()
Elliptic curves in class 414400.ck
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
414400.ck1 | 414400ck1 | \([0, 0, 0, -1688200, 844275000]\) | \(33256413948450816/2481997\) | \(39711952000000\) | \([2]\) | \(3133440\) | \(2.0590\) | \(\Gamma_0(N)\)-optimal |
414400.ck2 | 414400ck2 | \([0, 0, 0, -1684700, 847950000]\) | \(-2065624967846736/17960084863\) | \(-4597781724928000000\) | \([2]\) | \(6266880\) | \(2.4056\) |
Rank
sage: E.rank()
The elliptic curves in class 414400.ck have rank \(0\).
Complex multiplication
The elliptic curves in class 414400.ck do not have complex multiplication.Modular form 414400.2.a.ck
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.