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SageMath
E = EllipticCurve("v1")
E.isogeny_class()
Elliptic curves in class 8670v
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
8670.v6 | 8670v1 | \([1, 0, 0, -23126, 1661220]\) | \(-56667352321/16711680\) | \(-403379329105920\) | \([2]\) | \(36864\) | \(1.5171\) | \(\Gamma_0(N)\)-optimal |
8670.v5 | 8670v2 | \([1, 0, 0, -393046, 94807076]\) | \(278202094583041/16646400\) | \(401803628601600\) | \([2, 2]\) | \(73728\) | \(1.8636\) | |
8670.v4 | 8670v3 | \([1, 0, 0, -416166, 83020500]\) | \(330240275458561/67652010000\) | \(1632955059363690000\) | \([2, 2]\) | \(147456\) | \(2.2102\) | |
8670.v2 | 8670v4 | \([1, 0, 0, -6288646, 6069408116]\) | \(1139466686381936641/4080\) | \(98481281520\) | \([2]\) | \(147456\) | \(2.2102\) | |
8670.v3 | 8670v5 | \([1, 0, 0, -2086586, -1086607584]\) | \(41623544884956481/2962701562500\) | \(71512413391251562500\) | \([2, 2]\) | \(294912\) | \(2.5568\) | |
8670.v7 | 8670v6 | \([1, 0, 0, 884334, 498400200]\) | \(3168685387909439/6278181696900\) | \(-151540043903460836100\) | \([2]\) | \(294912\) | \(2.5568\) | |
8670.v1 | 8670v7 | \([1, 0, 0, -32792836, -72282118834]\) | \(161572377633716256481/914742821250\) | \(22079667965176541250\) | \([2]\) | \(589824\) | \(2.9033\) | |
8670.v8 | 8670v8 | \([1, 0, 0, 1892944, -4740612030]\) | \(31077313442863199/420227050781250\) | \(-10143259433898925781250\) | \([2]\) | \(589824\) | \(2.9033\) |
Rank
sage: E.rank()
The elliptic curves in class 8670v have rank \(1\).
Complex multiplication
The elliptic curves in class 8670v do not have complex multiplication.Modular form 8670.2.a.v
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.