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SageMath
E = EllipticCurve("do1")
E.isogeny_class()
Elliptic curves in class 139650.do
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
139650.do1 | 139650gs3 | \([1, 0, 1, -524326, 146088548]\) | \(8671983378625/82308\) | \(151303967062500\) | \([2]\) | \(1492992\) | \(1.8832\) | |
139650.do2 | 139650gs4 | \([1, 0, 1, -512076, 153242548]\) | \(-8078253774625/846825858\) | \(-1556690865122531250\) | \([2]\) | \(2985984\) | \(2.2297\) | |
139650.do3 | 139650gs1 | \([1, 0, 1, -9826, -29452]\) | \(57066625/32832\) | \(60353937000000\) | \([2]\) | \(497664\) | \(1.3339\) | \(\Gamma_0(N)\)-optimal |
139650.do4 | 139650gs2 | \([1, 0, 1, 39174, -225452]\) | \(3616805375/2105352\) | \(-3870196210125000\) | \([2]\) | \(995328\) | \(1.6804\) |
Rank
sage: E.rank()
The elliptic curves in class 139650.do have rank \(1\).
Complex multiplication
The elliptic curves in class 139650.do do not have complex multiplication.Modular form 139650.2.a.do
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.