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SageMath
E = EllipticCurve("k1")
E.isogeny_class()
Elliptic curves in class 324870.k
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
324870.k1 | 324870k3 | \([1, 1, 0, -25802298, 50436290388]\) | \(16147699491119193630361/273400196160\) | \(32165259678027840\) | \([2]\) | \(17915904\) | \(2.7099\) | |
324870.k2 | 324870k4 | \([1, 1, 0, -25776818, 50540906172]\) | \(-16099908724056356461081/66450404013575400\) | \(-7817823581793132234600\) | \([2]\) | \(35831808\) | \(3.0565\) | |
324870.k3 | 324870k1 | \([1, 1, 0, -338223, 60027633]\) | \(36370300595789161/7754268658500\) | \(912281953403866500\) | \([2]\) | \(5971968\) | \(2.1606\) | \(\Gamma_0(N)\)-optimal |
324870.k4 | 324870k2 | \([1, 1, 0, 738307, 365546847]\) | \(378307987602100919/708763802906250\) | \(-83385352648117406250\) | \([2]\) | \(11943936\) | \(2.5072\) |
Rank
sage: E.rank()
The elliptic curves in class 324870.k have rank \(1\).
Complex multiplication
The elliptic curves in class 324870.k do not have complex multiplication.Modular form 324870.2.a.k
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.