magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, -7, 56, 17, -246, -50, 554, 162, -708, -274, 555, 234, -274, -104, 83, 23, -14, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^19 - 2*x^18 - 14*x^17 + 23*x^16 + 83*x^15 - 104*x^14 - 274*x^13 + 234*x^12 + 555*x^11 - 274*x^10 - 708*x^9 + 162*x^8 + 554*x^7 - 50*x^6 - 246*x^5 + 17*x^4 + 56*x^3 - 7*x^2 - 5*x + 1)
gp: K = bnfinit(x^19 - 2*x^18 - 14*x^17 + 23*x^16 + 83*x^15 - 104*x^14 - 274*x^13 + 234*x^12 + 555*x^11 - 274*x^10 - 708*x^9 + 162*x^8 + 554*x^7 - 50*x^6 - 246*x^5 + 17*x^4 + 56*x^3 - 7*x^2 - 5*x + 1, 1)
Normalized defining polynomial
\( x^{19} - 2 x^{18} - 14 x^{17} + 23 x^{16} + 83 x^{15} - 104 x^{14} - 274 x^{13} + 234 x^{12} + 555 x^{11} - 274 x^{10} - 708 x^{9} + 162 x^{8} + 554 x^{7} - 50 x^{6} - 246 x^{5} + 17 x^{4} + 56 x^{3} - 7 x^{2} - 5 x + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $19$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[9, 5]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-1388142379667992699741943954318476=-\,2^{2}\cdot 347035594916998174935485988579619\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $55.51$ | 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: | $2, 347035594916998174935485988579619$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$
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: | $13$ | 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: | \( 8259829376.29 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$S_{19}$ (as 19T8):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A non-solvable group of order 121645100408832000 |
| The 490 conjugacy class representatives for $S_{19}$ are not computed |
| Character table for $S_{19}$ is not computed |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.9.0.1}{9} }$ | ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.7.0.1}{7} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ | $19$ | $18{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.7.0.1}{7} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ | $17{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | ${\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }^{2}$ | $17{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ | $18{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | ${\href{/LocalNumberField/37.14.0.1}{14} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.14.0.1}{14} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.13.0.1}{13} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }$ | $18{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.7.0.1}{7} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ | ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.0.1 | $x^{2} - x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 2.3.2.1 | $x^{3} - 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| 2.14.0.1 | $x^{14} - x^{5} - x^{3} - x + 1$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | |
| 347035594916998174935485988579619 | Data not computed | ||||||