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SageMath
E = EllipticCurve("ek1")
E.isogeny_class()
Elliptic curves in class 25200.ek
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25200.ek1 | 25200cw3 | \([0, 0, 0, -168075, 23240250]\) | \(416832723/56000\) | \(70543872000000000\) | \([2]\) | \(165888\) | \(1.9604\) | |
| 25200.ek2 | 25200cw1 | \([0, 0, 0, -42075, -3317750]\) | \(4767078987/6860\) | \(11854080000000\) | \([2]\) | \(55296\) | \(1.4111\) | \(\Gamma_0(N)\)-optimal |
| 25200.ek3 | 25200cw2 | \([0, 0, 0, -30075, -5249750]\) | \(-1740992427/5882450\) | \(-10164873600000000\) | \([2]\) | \(110592\) | \(1.7576\) | |
| 25200.ek4 | 25200cw4 | \([0, 0, 0, 263925, 123032250]\) | \(1613964717/6125000\) | \(-7715736000000000000\) | \([2]\) | \(331776\) | \(2.3069\) |
Rank
sage: E.rank()
The elliptic curves in class 25200.ek have rank \(1\).
Complex multiplication
The elliptic curves in class 25200.ek do not have complex multiplication.Modular form 25200.2.a.ek
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
