magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 4, -4, 10, -9, 16, -15, 21, -19, 23, -19, 21, -15, 16, -9, 10, -4, 4, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 4*x^18 - 4*x^17 + 10*x^16 - 9*x^15 + 16*x^14 - 15*x^13 + 21*x^12 - 19*x^11 + 23*x^10 - 19*x^9 + 21*x^8 - 15*x^7 + 16*x^6 - 9*x^5 + 10*x^4 - 4*x^3 + 4*x^2 - x + 1)
gp: K = bnfinit(x^20 - x^19 + 4*x^18 - 4*x^17 + 10*x^16 - 9*x^15 + 16*x^14 - 15*x^13 + 21*x^12 - 19*x^11 + 23*x^10 - 19*x^9 + 21*x^8 - 15*x^7 + 16*x^6 - 9*x^5 + 10*x^4 - 4*x^3 + 4*x^2 - x + 1, 1)
Normalized defining polynomial
\( x^{20} - x^{19} + 4 x^{18} - 4 x^{17} + 10 x^{16} - 9 x^{15} + 16 x^{14} - 15 x^{13} + 21 x^{12} - 19 x^{11} + 23 x^{10} - 19 x^{9} + 21 x^{8} - 15 x^{7} + 16 x^{6} - 9 x^{5} + 10 x^{4} - 4 x^{3} + 4 x^{2} - x + 1 \)
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: | \(2598700405016807521784377=31\cdot 223\cdot 16493^{2}\cdot 1175561^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $16.62$ | 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: | $31, 223, 16493, 1175561$ | 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: | \( -1 \) (order $2$) | 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: | \( 6156.49767397 \) (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 non-solvable group of order 3715891200 |
| The 481 conjugacy class representatives for t20n1110 are not computed |
| Character table for t20n1110 is not computed |
Intermediate fields
| 10.6.19388527573.1 |
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 | ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ | ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ | ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ | $16{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ | ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/29.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }$ | R | ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | $18{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ | ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $31$ | 31.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 31.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.2.1.2 | $x^{2} + 217$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.6.0.1 | $x^{6} - 2 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
| 31.8.0.1 | $x^{8} - x + 22$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 223 | Data not computed | ||||||
| 16493 | Data not computed | ||||||
| 1175561 | Data not computed | ||||||