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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 86190.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
86190.a1 | 86190i4 | \([1, 1, 0, -11592558, 15183418398]\) | \(35694515311673154481/10400566692750\) | \(50201548917665934750\) | \([2]\) | \(5677056\) | \(2.7597\) | |
86190.a2 | 86190i3 | \([1, 1, 0, -5674178, -5081264118]\) | \(4185743240664514801/113629394531250\) | \(548467384187988281250\) | \([2]\) | \(5677056\) | \(2.7597\) | |
86190.a3 | 86190i2 | \([1, 1, 0, -818808, 171275148]\) | \(12577973014374481/4642947562500\) | \(22410621081203062500\) | \([2, 2]\) | \(2838528\) | \(2.4131\) | |
86190.a4 | 86190i1 | \([1, 1, 0, 158012, 19086592]\) | \(90391899763439/84690294000\) | \(-408783873291846000\) | \([2]\) | \(1419264\) | \(2.0665\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 86190.a have rank \(0\).
Complex multiplication
The elliptic curves in class 86190.a do not have complex multiplication.Modular form 86190.2.a.a
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.