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SageMath
E = EllipticCurve("er1")
E.isogeny_class()
Elliptic curves in class 55440er
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
55440.dw3 | 55440er1 | \([0, 0, 0, -142707, -14301646]\) | \(107639597521009/32699842560\) | \(97641206686679040\) | \([2]\) | \(491520\) | \(1.9650\) | \(\Gamma_0(N)\)-optimal |
55440.dw2 | 55440er2 | \([0, 0, 0, -879987, 306710066]\) | \(25238585142450289/995844326400\) | \(2973575225121177600\) | \([2, 2]\) | \(983040\) | \(2.3116\) | |
55440.dw4 | 55440er3 | \([0, 0, 0, 387213, 1117464626]\) | \(2150235484224911/181905111732960\) | \(-543165753152830832640\) | \([2]\) | \(1966080\) | \(2.6581\) | |
55440.dw1 | 55440er4 | \([0, 0, 0, -13943667, 20040705074]\) | \(100407751863770656369/166028940000\) | \(495759758376960000\) | \([2]\) | \(1966080\) | \(2.6581\) |
Rank
sage: E.rank()
The elliptic curves in class 55440er have rank \(1\).
Complex multiplication
The elliptic curves in class 55440er do not have complex multiplication.Modular form 55440.2.a.er
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 Cremona 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 Cremona labels.