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SageMath
E = EllipticCurve("ck1")
E.isogeny_class()
Elliptic curves in class 324870.ck
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
324870.ck1 | 324870ck3 | \([1, 0, 1, -2013583, 966826298]\) | \(7674388308884766169/1007648705929320\) | \(118548862603878568680\) | \([2]\) | \(14155776\) | \(2.5800\) | |
324870.ck2 | 324870ck2 | \([1, 0, 1, -514183, -126536182]\) | \(127787213284071769/15197834433600\) | \(1788010023278606400\) | \([2, 2]\) | \(7077888\) | \(2.2334\) | |
324870.ck3 | 324870ck1 | \([1, 0, 1, -498503, -135511414]\) | \(116449478628435289/1996001280\) | \(234827554590720\) | \([2]\) | \(3538944\) | \(1.8868\) | \(\Gamma_0(N)\)-optimal |
324870.ck4 | 324870ck4 | \([1, 0, 1, 734337, -645421094]\) | \(372239584720800551/1745320379985000\) | \(-205335197384855265000\) | \([2]\) | \(14155776\) | \(2.5800\) |
Rank
sage: E.rank()
The elliptic curves in class 324870.ck have rank \(1\).
Complex multiplication
The elliptic curves in class 324870.ck do not have complex multiplication.Modular form 324870.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.