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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 510.e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
510.e1 | 510e7 | \([1, 1, 1, -113470, -14759143]\) | \(161572377633716256481/914742821250\) | \(914742821250\) | \([2]\) | \(2048\) | \(1.4867\) | |
510.e2 | 510e4 | \([1, 1, 1, -21760, 1226417]\) | \(1139466686381936641/4080\) | \(4080\) | \([4]\) | \(512\) | \(0.79359\) | |
510.e3 | 510e5 | \([1, 1, 1, -7220, -224143]\) | \(41623544884956481/2962701562500\) | \(2962701562500\) | \([2, 2]\) | \(1024\) | \(1.1402\) | |
510.e4 | 510e3 | \([1, 1, 1, -1440, 16305]\) | \(330240275458561/67652010000\) | \(67652010000\) | \([2, 4]\) | \(512\) | \(0.79359\) | |
510.e5 | 510e2 | \([1, 1, 1, -1360, 18737]\) | \(278202094583041/16646400\) | \(16646400\) | \([2, 4]\) | \(256\) | \(0.44702\) | |
510.e6 | 510e1 | \([1, 1, 1, -80, 305]\) | \(-56667352321/16711680\) | \(-16711680\) | \([4]\) | \(128\) | \(0.10045\) | \(\Gamma_0(N)\)-optimal |
510.e7 | 510e6 | \([1, 1, 1, 3060, 102705]\) | \(3168685387909439/6278181696900\) | \(-6278181696900\) | \([4]\) | \(1024\) | \(1.1402\) | |
510.e8 | 510e8 | \([1, 1, 1, 6550, -962215]\) | \(31077313442863199/420227050781250\) | \(-420227050781250\) | \([2]\) | \(2048\) | \(1.4867\) |
Rank
sage: E.rank()
The elliptic curves in class 510.e have rank \(0\).
Complex multiplication
The elliptic curves in class 510.e do not have complex multiplication.Modular form 510.2.a.e
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.