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Properties

Label 22.12.8719272932...7207.2
Degree $22$
Signature $[12, 5]$
Discriminant $-\,23^{21}\cdot 47^{2}$
Root discriminant $28.30$
Ramified primes $23, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T28

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-229, -560, 3353, -2191, -6254, 9989, 174, -11765, 8353, 4752, -8668, 574, 4746, -1536, -1857, 1079, 384, -404, 12, 65, -14, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 3*x^21 - 14*x^20 + 65*x^19 + 12*x^18 - 404*x^17 + 384*x^16 + 1079*x^15 - 1857*x^14 - 1536*x^13 + 4746*x^12 + 574*x^11 - 8668*x^10 + 4752*x^9 + 8353*x^8 - 11765*x^7 + 174*x^6 + 9989*x^5 - 6254*x^4 - 2191*x^3 + 3353*x^2 - 560*x - 229)
 
gp: K = bnfinit(x^22 - 3*x^21 - 14*x^20 + 65*x^19 + 12*x^18 - 404*x^17 + 384*x^16 + 1079*x^15 - 1857*x^14 - 1536*x^13 + 4746*x^12 + 574*x^11 - 8668*x^10 + 4752*x^9 + 8353*x^8 - 11765*x^7 + 174*x^6 + 9989*x^5 - 6254*x^4 - 2191*x^3 + 3353*x^2 - 560*x - 229, 1)
 

Normalized defining polynomial

\( x^{22} - 3 x^{21} - 14 x^{20} + 65 x^{19} + 12 x^{18} - 404 x^{17} + 384 x^{16} + 1079 x^{15} - 1857 x^{14} - 1536 x^{13} + 4746 x^{12} + 574 x^{11} - 8668 x^{10} + 4752 x^{9} + 8353 x^{8} - 11765 x^{7} + 174 x^{6} + 9989 x^{5} - 6254 x^{4} - 2191 x^{3} + 3353 x^{2} - 560 x - 229 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-87192729322616328324934343477207=-\,23^{21}\cdot 47^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.30$
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:  $23, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{139} a^{19} - \frac{3}{139} a^{18} - \frac{52}{139} a^{17} + \frac{5}{139} a^{16} + \frac{8}{139} a^{15} - \frac{25}{139} a^{14} + \frac{44}{139} a^{13} - \frac{58}{139} a^{12} - \frac{13}{139} a^{11} - \frac{38}{139} a^{10} + \frac{42}{139} a^{9} - \frac{29}{139} a^{8} - \frac{38}{139} a^{7} - \frac{64}{139} a^{6} - \frac{30}{139} a^{5} + \frac{59}{139} a^{4} - \frac{60}{139} a^{3} + \frac{40}{139} a^{2} - \frac{62}{139} a + \frac{57}{139}$, $\frac{1}{139} a^{20} - \frac{61}{139} a^{18} - \frac{12}{139} a^{17} + \frac{23}{139} a^{16} - \frac{1}{139} a^{15} - \frac{31}{139} a^{14} - \frac{65}{139} a^{13} - \frac{48}{139} a^{12} + \frac{62}{139} a^{11} + \frac{67}{139} a^{10} - \frac{42}{139} a^{9} + \frac{14}{139} a^{8} - \frac{39}{139} a^{7} + \frac{56}{139} a^{6} - \frac{31}{139} a^{5} - \frac{22}{139} a^{4} - \frac{1}{139} a^{3} + \frac{58}{139} a^{2} + \frac{10}{139} a + \frac{32}{139}$, $\frac{1}{4387795647221316037} a^{21} - \frac{4505905196065439}{4387795647221316037} a^{20} + \frac{4109227079674462}{4387795647221316037} a^{19} + \frac{681421437249097290}{4387795647221316037} a^{18} - \frac{691077048731177275}{4387795647221316037} a^{17} + \frac{1592306835069118989}{4387795647221316037} a^{16} - \frac{1724743927135196475}{4387795647221316037} a^{15} - \frac{926326814243294068}{4387795647221316037} a^{14} + \frac{675396393919730101}{4387795647221316037} a^{13} - \frac{15037216727479720}{31566875159865583} a^{12} + \frac{1599832074635627232}{4387795647221316037} a^{11} + \frac{417222827585831901}{4387795647221316037} a^{10} - \frac{822231638037448757}{4387795647221316037} a^{9} + \frac{1643789409656093837}{4387795647221316037} a^{8} - \frac{1385754765853519677}{4387795647221316037} a^{7} - \frac{667861865527123903}{4387795647221316037} a^{6} + \frac{2051283390731810091}{4387795647221316037} a^{5} + \frac{1441004510691661679}{4387795647221316037} a^{4} - \frac{2157190647016781896}{4387795647221316037} a^{3} - \frac{396139687625260275}{4387795647221316037} a^{2} + \frac{2190909682911474118}{4387795647221316037} a - \frac{5363056605593242}{19160679682189153}$

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:  $16$
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:  \( 73339658.3874 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T28:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 22528
The 208 conjugacy class representatives for t22n28 are not computed
Character table for t22n28 is not computed

Intermediate fields

\(\Q(\zeta_{23})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 22 siblings: data not computed
Degree 44 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/3.11.0.1}{11} }^{2}$ $22$ $22$ $22$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ $22$ $22$ R ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/41.11.0.1}{11} }^{2}$ $22$ R $22$ ${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
23Data not computed
$47$$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{47}$$x + 2$$1$$1$$0$Trivial$[\ ]$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.0.1$x^{2} - x + 13$$1$$2$$0$$C_2$$[\ ]^{2}$
47.2.1.2$x^{2} + 94$$2$$1$$1$$C_2$$[\ ]_{2}$