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SageMath
E = EllipticCurve("cj1")
E.isogeny_class()
Elliptic curves in class 182070cj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
182070.bi7 | 182070cj1 | \([1, -1, 0, 546156, -117201200]\) | \(1023887723039/928972800\) | \(-16346472748100812800\) | \([2]\) | \(5242880\) | \(2.3740\) | \(\Gamma_0(N)\)-optimal |
182070.bi6 | 182070cj2 | \([1, -1, 0, -2783124, -1047402032]\) | \(135487869158881/51438240000\) | \(905122075016910240000\) | \([2, 2]\) | \(10485760\) | \(2.7206\) | |
182070.bi5 | 182070cj3 | \([1, -1, 0, -19637604, 32752572160]\) | \(47595748626367201/1215506250000\) | \(21388397798914256250000\) | \([2, 2]\) | \(20971520\) | \(3.0672\) | |
182070.bi4 | 182070cj4 | \([1, -1, 0, -39197124, -94420180832]\) | \(378499465220294881/120530818800\) | \(2120894976494981458800\) | \([2]\) | \(20971520\) | \(3.0672\) | |
182070.bi2 | 182070cj5 | \([1, -1, 0, -312250104, 2123820019660]\) | \(191342053882402567201/129708022500\) | \(2282379694008583522500\) | \([2, 2]\) | \(41943040\) | \(3.4138\) | |
182070.bi8 | 182070cj6 | \([1, -1, 0, 3303216, 104681219188]\) | \(226523624554079/269165039062500\) | \(-4736305493311157226562500\) | \([2]\) | \(41943040\) | \(3.4138\) | |
182070.bi1 | 182070cj7 | \([1, -1, 0, -4996000854, 135920780694310]\) | \(783736670177727068275201/360150\) | \(6337303051530150\) | \([2]\) | \(83886080\) | \(3.7603\) | |
182070.bi3 | 182070cj8 | \([1, -1, 0, -310299354, 2151664635010]\) | \(-187778242790732059201/4984939585440150\) | \(-87716431616002508660610150\) | \([2]\) | \(83886080\) | \(3.7603\) |
Rank
sage: E.rank()
The elliptic curves in class 182070cj have rank \(2\).
Complex multiplication
The elliptic curves in class 182070cj do not have complex multiplication.Modular form 182070.2.a.cj
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.