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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 463680.bn
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
463680.bn1 | 463680bn7 | \([0, 0, 0, -146245833228, -20197303934266352]\) | \(1810117493172631097464564372609/125368453502655029296875000\) | \(23958284560875000000000000000000000\) | \([2]\) | \(3057647616\) | \(5.3448\) | |
463680.bn2 | 463680bn6 | \([0, 0, 0, -143722884108, -20971765552195568]\) | \(1718043013877225552292911401729/9180538178765625000000\) | \(1754428167243730944000000000000\) | \([2, 2]\) | \(1528823808\) | \(4.9982\) | |
463680.bn3 | 463680bn3 | \([0, 0, 0, -143722699788, -20971822033225712]\) | \(1718036403880129446396978632449/49057344000000\) | \(9375004433055744000000\) | \([2]\) | \(764411904\) | \(4.6516\) | |
463680.bn4 | 463680bn8 | \([0, 0, 0, -141202884108, -21742612384195568]\) | \(-1629247127728109256861881401729/125809119536174660320875000\) | \(-24042497151302717243108327424000000\) | \([2]\) | \(3057647616\) | \(5.3448\) | |
463680.bn5 | 463680bn4 | \([0, 0, 0, -27254578188, 1725900003588112]\) | \(11715873038622856702991202049/46415372499833400000000\) | \(8870115816866722244198400000000\) | \([2]\) | \(1019215872\) | \(4.7955\) | \(\Gamma_0(N)\)-optimal* |
463680.bn6 | 463680bn2 | \([0, 0, 0, -2530630668, -1938013636592]\) | \(9378698233516887309850369/5418996968417034240000\) | \(1035586447599473252357898240000\) | \([2, 2]\) | \(509607936\) | \(4.4489\) | \(\Gamma_0(N)\)-optimal* |
463680.bn7 | 463680bn1 | \([0, 0, 0, -1775655948, -28723610732528]\) | \(3239908336204082689644289/9880281924658790400\) | \(1888151279521302630000230400\) | \([2]\) | \(254803968\) | \(4.1023\) | \(\Gamma_0(N)\)-optimal* |
463680.bn8 | 463680bn5 | \([0, 0, 0, 10113721332, -15497816721392]\) | \(598672364899527954087397631/346996861747253448998400\) | \(-66312132942560693929858459238400\) | \([2]\) | \(1019215872\) | \(4.7955\) |
Rank
sage: E.rank()
The elliptic curves in class 463680.bn have rank \(1\).
Complex multiplication
The elliptic curves in class 463680.bn do not have complex multiplication.Modular form 463680.2.a.bn
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 3 & 6 & 12 & 12 \\ 2 & 1 & 2 & 2 & 6 & 3 & 6 & 6 \\ 4 & 2 & 1 & 4 & 12 & 6 & 3 & 12 \\ 4 & 2 & 4 & 1 & 12 & 6 & 12 & 3 \\ 3 & 6 & 12 & 12 & 1 & 2 & 4 & 4 \\ 6 & 3 & 6 & 6 & 2 & 1 & 2 & 2 \\ 12 & 6 & 3 & 12 & 4 & 2 & 1 & 4 \\ 12 & 6 & 12 & 3 & 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.