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SageMath
E = EllipticCurve("bu1")
E.isogeny_class()
Elliptic curves in class 130050bu
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 130050.fr6 | 130050bu1 | \([1, -1, 1, -5203355, -5611820853]\) | \(-56667352321/16711680\) | \(-4594742670597120000000\) | \([4]\) | \(7077888\) | \(2.8711\) | \(\Gamma_0(N)\)-optimal |
| 130050.fr5 | 130050bu2 | \([1, -1, 1, -88435355, -320062316853]\) | \(278202094583041/16646400\) | \(4576794457040100000000\) | \([2, 2]\) | \(14155776\) | \(3.2177\) | |
| 130050.fr4 | 130050bu3 | \([1, -1, 1, -93637355, -280287824853]\) | \(330240275458561/67652010000\) | \(18600378723064531406250000\) | \([2, 2]\) | \(28311552\) | \(3.5642\) | |
| 130050.fr2 | 130050bu4 | \([1, -1, 1, -1414945355, -20485667336853]\) | \(1139466686381936641/4080\) | \(1121763347313750000\) | \([2]\) | \(28311552\) | \(3.5642\) | |
| 130050.fr3 | 130050bu5 | \([1, -1, 1, -469481855, 3666831114147]\) | \(41623544884956481/2962701562500\) | \(814571083784724829101562500\) | \([2, 2]\) | \(56623104\) | \(3.9108\) | |
| 130050.fr7 | 130050bu6 | \([1, -1, 1, 198975145, -1681901699853]\) | \(3168685387909439/6278181696900\) | \(-1726135812587858586201562500\) | \([2]\) | \(56623104\) | \(3.9108\) | |
| 130050.fr1 | 130050bu7 | \([1, -1, 1, -7378388105, 243944772676647]\) | \(161572377633716256481/914742821250\) | \(251501217915839040175781250\) | \([2]\) | \(113246208\) | \(4.2574\) | |
| 130050.fr8 | 130050bu8 | \([1, -1, 1, 425912395, 15999991513647]\) | \(31077313442863199/420227050781250\) | \(-115538064489254951477050781250\) | \([2]\) | \(113246208\) | \(4.2574\) |
Rank
sage: E.rank()
The elliptic curves in class 130050bu have rank \(1\).
Complex multiplication
The elliptic curves in class 130050bu do not have complex multiplication.Modular form 130050.2.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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
