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SageMath
E = EllipticCurve("bz1")
E.isogeny_class()
Elliptic curves in class 47610bz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
47610.bu8 | 47610bz1 | \([1, -1, 1, 7042, 696701]\) | \(357911/2160\) | \(-233103232254960\) | \([2]\) | \(202752\) | \(1.4388\) | \(\Gamma_0(N)\)-optimal |
47610.bu6 | 47610bz2 | \([1, -1, 1, -88178, 9152237]\) | \(702595369/72900\) | \(7867234088604900\) | \([2, 2]\) | \(405504\) | \(1.7853\) | |
47610.bu7 | 47610bz3 | \([1, -1, 1, -64373, -20556403]\) | \(-273359449/1536000\) | \(-165762298492416000\) | \([2]\) | \(608256\) | \(1.9881\) | |
47610.bu5 | 47610bz4 | \([1, -1, 1, -326228, -61691443]\) | \(35578826569/5314410\) | \(573521365059297210\) | \([2]\) | \(811008\) | \(2.1319\) | |
47610.bu4 | 47610bz5 | \([1, -1, 1, -1373648, 620007581]\) | \(2656166199049/33750\) | \(3642238003983750\) | \([2]\) | \(811008\) | \(2.1319\) | |
47610.bu3 | 47610bz6 | \([1, -1, 1, -1587893, -768300019]\) | \(4102915888729/9000000\) | \(971263467729000000\) | \([2, 2]\) | \(1216512\) | \(2.3346\) | |
47610.bu1 | 47610bz7 | \([1, -1, 1, -25392893, -49244802019]\) | \(16778985534208729/81000\) | \(8741371209561000\) | \([2]\) | \(2433024\) | \(2.6812\) | |
47610.bu2 | 47610bz8 | \([1, -1, 1, -2159213, -165671683]\) | \(10316097499609/5859375000\) | \(632332986802734375000\) | \([2]\) | \(2433024\) | \(2.6812\) |
Rank
sage: E.rank()
The elliptic curves in class 47610bz have rank \(0\).
Complex multiplication
The elliptic curves in class 47610bz do not have complex multiplication.Modular form 47610.2.a.bz
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.