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SageMath
E = EllipticCurve("bu1")
E.isogeny_class()
Elliptic curves in class 45570bu
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 45570.bo6 | 45570bu1 | \([1, 1, 1, 2939, 348683]\) | \(23862997439/457113600\) | \(-53778957926400\) | \([2]\) | \(196608\) | \(1.3151\) | \(\Gamma_0(N)\)-optimal |
| 45570.bo5 | 45570bu2 | \([1, 1, 1, -59781, 5291019]\) | \(200828550012481/12454560000\) | \(1465266529440000\) | \([2, 2]\) | \(393216\) | \(1.6616\) | |
| 45570.bo4 | 45570bu3 | \([1, 1, 1, -181301, -23241877]\) | \(5601911201812801/1271193750000\) | \(149554673493750000\) | \([2]\) | \(786432\) | \(2.0082\) | |
| 45570.bo2 | 45570bu4 | \([1, 1, 1, -941781, 351387819]\) | \(785209010066844481/3324675600\) | \(391144759664400\) | \([2, 2]\) | \(786432\) | \(2.0082\) | |
| 45570.bo3 | 45570bu5 | \([1, 1, 1, -927081, 362906739]\) | \(-749011598724977281/51173462246460\) | \(-6020506659833772540\) | \([2]\) | \(1572864\) | \(2.3548\) | |
| 45570.bo1 | 45570bu6 | \([1, 1, 1, -15068481, 22507704099]\) | \(3216206300355197383681/57660\) | \(6783641340\) | \([2]\) | \(1572864\) | \(2.3548\) |
Rank
sage: E.rank()
The elliptic curves in class 45570bu have rank \(2\).
Complex multiplication
The elliptic curves in class 45570bu do not have complex multiplication.Modular form 45570.2.a.bu
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
