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SageMath
E = EllipticCurve("bo1")
E.isogeny_class()
Elliptic curves in class 189525.bo
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
189525.bo1 | 189525bi3 | \([1, 0, 1, -1015501, 393771023]\) | \(157551496201/13125\) | \(9648081064453125\) | \([2]\) | \(2322432\) | \(2.1110\) | |
189525.bo2 | 189525bi2 | \([1, 0, 1, -67876, 5244773]\) | \(47045881/11025\) | \(8104388094140625\) | \([2, 2]\) | \(1161216\) | \(1.7644\) | |
189525.bo3 | 189525bi1 | \([1, 0, 1, -22751, -1253227]\) | \(1771561/105\) | \(77184648515625\) | \([2]\) | \(580608\) | \(1.4178\) | \(\Gamma_0(N)\)-optimal |
189525.bo4 | 189525bi4 | \([1, 0, 1, 157749, 32771023]\) | \(590589719/972405\) | \(-714807029903203125\) | \([2]\) | \(2322432\) | \(2.1110\) |
Rank
sage: E.rank()
The elliptic curves in class 189525.bo have rank \(1\).
Complex multiplication
The elliptic curves in class 189525.bo do not have complex multiplication.Modular form 189525.2.a.bo
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.