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SageMath
E = EllipticCurve("ck1")
E.isogeny_class()
Elliptic curves in class 125840ck
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
125840.cp2 | 125840ck1 | \([0, -1, 0, -62960, 6065600]\) | \(3803721481/26000\) | \(188664160256000\) | \([2]\) | \(622080\) | \(1.5732\) | \(\Gamma_0(N)\)-optimal |
125840.cp3 | 125840ck2 | \([0, -1, 0, -24240, 13406912]\) | \(-217081801/10562500\) | \(-76644815104000000\) | \([2]\) | \(1244160\) | \(1.9198\) | |
125840.cp1 | 125840ck3 | \([0, -1, 0, -401760, -93948160]\) | \(988345570681/44994560\) | \(326494649172623360\) | \([2]\) | \(1866240\) | \(2.1225\) | |
125840.cp4 | 125840ck4 | \([0, -1, 0, 217760, -358111488]\) | \(157376536199/7722894400\) | \(-56039745643144806400\) | \([2]\) | \(3732480\) | \(2.4691\) |
Rank
sage: E.rank()
The elliptic curves in class 125840ck have rank \(0\).
Complex multiplication
The elliptic curves in class 125840ck do not have complex multiplication.Modular form 125840.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.