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SageMath
E = EllipticCurve("eg1")
E.isogeny_class()
Elliptic curves in class 169050.eg
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
169050.eg1 | 169050gv3 | \([1, 0, 1, -219090051, 1248175760248]\) | \(632678989847546725777/80515134\) | \(148008203124468750\) | \([2]\) | \(23592960\) | \(3.1547\) | |
169050.eg2 | 169050gv4 | \([1, 0, 1, -15666551, 13514577248]\) | \(231331938231569617/90942310746882\) | \(167176123704061256531250\) | \([2]\) | \(23592960\) | \(3.1547\) | |
169050.eg3 | 169050gv2 | \([1, 0, 1, -13694301, 19498383748]\) | \(154502321244119857/55101928644\) | \(101291981297468062500\) | \([2, 2]\) | \(11796480\) | \(2.8081\) | |
169050.eg4 | 169050gv1 | \([1, 0, 1, -733801, 394606748]\) | \(-23771111713777/22848457968\) | \(-42001534866831750000\) | \([2]\) | \(5898240\) | \(2.4615\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 169050.eg have rank \(1\).
Complex multiplication
The elliptic curves in class 169050.eg do not have complex multiplication.Modular form 169050.2.a.eg
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.