magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, 46, -129, 247, -344, 365, -309, 221, -143, 89, -55, 34, -21, 13, -8, 5, -3, 2, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 2*x^18 - 3*x^17 + 5*x^16 - 8*x^15 + 13*x^14 - 21*x^13 + 34*x^12 - 55*x^11 + 89*x^10 - 143*x^9 + 221*x^8 - 309*x^7 + 365*x^6 - 344*x^5 + 247*x^4 - 129*x^3 + 46*x^2 - 10*x + 1)
gp: K = bnfinit(x^20 - x^19 + 2*x^18 - 3*x^17 + 5*x^16 - 8*x^15 + 13*x^14 - 21*x^13 + 34*x^12 - 55*x^11 + 89*x^10 - 143*x^9 + 221*x^8 - 309*x^7 + 365*x^6 - 344*x^5 + 247*x^4 - 129*x^3 + 46*x^2 - 10*x + 1, 1)
Normalized defining polynomial
\( x^{20} - x^{19} + 2 x^{18} - 3 x^{17} + 5 x^{16} - 8 x^{15} + 13 x^{14} - 21 x^{13} + 34 x^{12} - 55 x^{11} + 89 x^{10} - 143 x^{9} + 221 x^{8} - 309 x^{7} + 365 x^{6} - 344 x^{5} + 247 x^{4} - 129 x^{3} + 46 x^{2} - 10 x + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $20$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 10]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(4663997797513406792383141=11^{18}\cdot 397\cdot 2113\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $17.12$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $11, 397, 2113$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp
Unit group
magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
| Rank: | $9$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -89 a^{19} + 13 a^{18} - 166 a^{17} + 126 a^{16} - 335 a^{15} + 427 a^{14} - 788 a^{13} + 1196 a^{12} - 1997 a^{11} + 3185 a^{10} - 5186 a^{9} + 8281 a^{8} - 12563 a^{7} + 16723 a^{6} - 18128 a^{5} + 15042 a^{4} - 9048 a^{3} + 3692 a^{2} - 912 a + 103 \) (order $22$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 128094.732594 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 10240 |
| The 136 conjugacy class representatives for t20n409 are not computed |
| Character table for t20n409 is not computed |
Intermediate fields
| \(\Q(\sqrt{-11}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{11})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $20$ | ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ | R | ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{10}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
magma: idealfactors := Factorization(p*Integers(K)); // get the data
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
gp: idealfactors = idealprimedec(K, p); \\ get the data
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $11$ | 11.10.9.7 | $x^{10} + 2673$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
| 11.10.9.7 | $x^{10} + 2673$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ | |
| 397 | Data not computed | ||||||
| 2113 | Data not computed | ||||||