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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 184110.w
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
184110.w1 | 184110bc7 | \([1, 0, 1, -40962678, 100905259198]\) | \(161572377633716256481/914742821250\) | \(43034881914131771250\) | \([2]\) | \(14155776\) | \(2.9590\) | |
184110.w2 | 184110bc4 | \([1, 0, 1, -7855368, -8474838362]\) | \(1139466686381936641/4080\) | \(191947194480\) | \([2]\) | \(3538944\) | \(2.2658\) | |
184110.w3 | 184110bc5 | \([1, 0, 1, -2606428, 1516544198]\) | \(41623544884956481/2962701562500\) | \(139382905147889062500\) | \([2, 2]\) | \(7077888\) | \(2.6124\) | |
184110.w4 | 184110bc3 | \([1, 0, 1, -519848, -115995994]\) | \(330240275458561/67652010000\) | \(3182748411870810000\) | \([2, 2]\) | \(3538944\) | \(2.2658\) | |
184110.w5 | 184110bc2 | \([1, 0, 1, -490968, -132446042]\) | \(278202094583041/16646400\) | \(783144553478400\) | \([2, 2]\) | \(1769472\) | \(1.9192\) | |
184110.w6 | 184110bc1 | \([1, 0, 1, -28888, -2324314]\) | \(-56667352321/16711680\) | \(-786215708590080\) | \([2]\) | \(884736\) | \(1.5727\) | \(\Gamma_0(N)\)-optimal |
184110.w7 | 184110bc6 | \([1, 0, 1, 1104652, -695617594]\) | \(3168685387909439/6278181696900\) | \(-295362589008735468900\) | \([2]\) | \(7077888\) | \(2.6124\) | |
184110.w8 | 184110bc8 | \([1, 0, 1, 2364542, 6618747806]\) | \(31077313442863199/420227050781250\) | \(-19769951824035644531250\) | \([2]\) | \(14155776\) | \(2.9590\) |
Rank
sage: E.rank()
The elliptic curves in class 184110.w have rank \(1\).
Complex multiplication
The elliptic curves in class 184110.w do not have complex multiplication.Modular form 184110.2.a.w
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 16 & 2 & 4 & 8 & 16 & 8 & 4 \\ 16 & 1 & 8 & 4 & 2 & 4 & 8 & 16 \\ 2 & 8 & 1 & 2 & 4 & 8 & 4 & 2 \\ 4 & 4 & 2 & 1 & 2 & 4 & 2 & 4 \\ 8 & 2 & 4 & 2 & 1 & 2 & 4 & 8 \\ 16 & 4 & 8 & 4 & 2 & 1 & 8 & 16 \\ 8 & 8 & 4 & 2 & 4 & 8 & 1 & 8 \\ 4 & 16 & 2 & 4 & 8 & 16 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.