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SageMath
E = EllipticCurve("ck1")
E.isogeny_class()
Elliptic curves in class 166410.ck
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
166410.ck1 | 166410t1 | \([1, -1, 1, -3178778, 2012065881]\) | \(770842973809/66873600\) | \(308171849611339065600\) | \([2]\) | \(9461760\) | \(2.6722\) | \(\Gamma_0(N)\)-optimal |
166410.ck2 | 166410t2 | \([1, -1, 1, 3477622, 9323455641]\) | \(1009328859791/8734528080\) | \(-40251095707361023705680\) | \([2]\) | \(18923520\) | \(3.0188\) |
Rank
sage: E.rank()
The elliptic curves in class 166410.ck have rank \(0\).
Complex multiplication
The elliptic curves in class 166410.ck do not have complex multiplication.Modular form 166410.2.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.