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SageMath
E = EllipticCurve("bo1")
E.isogeny_class()
Elliptic curves in class 265200.bo
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
265200.bo1 | 265200bo5 | \([0, -1, 0, -8069608, 8824545712]\) | \(908031902324522977/161726530797\) | \(10350497971008000000\) | \([2]\) | \(8388608\) | \(2.6530\) | |
265200.bo2 | 265200bo3 | \([0, -1, 0, -555608, 108305712]\) | \(296380748763217/92608836489\) | \(5926965535296000000\) | \([2, 2]\) | \(4194304\) | \(2.3064\) | |
265200.bo3 | 265200bo2 | \([0, -1, 0, -217608, -37710288]\) | \(17806161424897/668584449\) | \(42789404736000000\) | \([2, 2]\) | \(2097152\) | \(1.9599\) | |
265200.bo4 | 265200bo1 | \([0, -1, 0, -215608, -38462288]\) | \(17319700013617/25857\) | \(1654848000000\) | \([2]\) | \(1048576\) | \(1.6133\) | \(\Gamma_0(N)\)-optimal |
265200.bo5 | 265200bo4 | \([0, -1, 0, 88392, -135630288]\) | \(1193377118543/124806800313\) | \(-7987635220032000000\) | \([2]\) | \(4194304\) | \(2.3064\) | |
265200.bo6 | 265200bo6 | \([0, -1, 0, 1550392, 731681712]\) | \(6439735268725823/7345472585373\) | \(-470110245463872000000\) | \([2]\) | \(8388608\) | \(2.6530\) |
Rank
sage: E.rank()
The elliptic curves in class 265200.bo have rank \(0\).
Complex multiplication
The elliptic curves in class 265200.bo do not have complex multiplication.Modular form 265200.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.