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Properties

Label 24.24.5025907025...7377.1
Degree $24$
Signature $[24, 0]$
Discriminant $17^{21}\cdot 31^{16}$
Root discriminant $117.73$
Ramified primes $17, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{24}$ (as 24T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-21745039, 49495809, 269369498, -273937411, -731825889, 619882563, 857890224, -726080555, -508719376, 477755253, 153613396, -182664671, -19336098, 41207184, -779357, -5531777, 513767, 441083, -61013, -20333, 3404, 498, -93, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 5*x^23 - 93*x^22 + 498*x^21 + 3404*x^20 - 20333*x^19 - 61013*x^18 + 441083*x^17 + 513767*x^16 - 5531777*x^15 - 779357*x^14 + 41207184*x^13 - 19336098*x^12 - 182664671*x^11 + 153613396*x^10 + 477755253*x^9 - 508719376*x^8 - 726080555*x^7 + 857890224*x^6 + 619882563*x^5 - 731825889*x^4 - 273937411*x^3 + 269369498*x^2 + 49495809*x - 21745039)
 
gp: K = bnfinit(x^24 - 5*x^23 - 93*x^22 + 498*x^21 + 3404*x^20 - 20333*x^19 - 61013*x^18 + 441083*x^17 + 513767*x^16 - 5531777*x^15 - 779357*x^14 + 41207184*x^13 - 19336098*x^12 - 182664671*x^11 + 153613396*x^10 + 477755253*x^9 - 508719376*x^8 - 726080555*x^7 + 857890224*x^6 + 619882563*x^5 - 731825889*x^4 - 273937411*x^3 + 269369498*x^2 + 49495809*x - 21745039, 1)
 

Normalized defining polynomial

\( x^{24} - 5 x^{23} - 93 x^{22} + 498 x^{21} + 3404 x^{20} - 20333 x^{19} - 61013 x^{18} + 441083 x^{17} + 513767 x^{16} - 5531777 x^{15} - 779357 x^{14} + 41207184 x^{13} - 19336098 x^{12} - 182664671 x^{11} + 153613396 x^{10} + 477755253 x^{9} - 508719376 x^{8} - 726080555 x^{7} + 857890224 x^{6} + 619882563 x^{5} - 731825889 x^{4} - 273937411 x^{3} + 269369498 x^{2} + 49495809 x - 21745039 \)

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

Invariants

Degree:  $24$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[24, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(50259070254551029841638058950925276817543684567377=17^{21}\cdot 31^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $117.73$
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:  $17, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(527=17\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{527}(1,·)$, $\chi_{527}(67,·)$, $\chi_{527}(373,·)$, $\chi_{527}(94,·)$, $\chi_{527}(273,·)$, $\chi_{527}(404,·)$, $\chi_{527}(149,·)$, $\chi_{527}(342,·)$, $\chi_{527}(87,·)$, $\chi_{527}(280,·)$, $\chi_{527}(25,·)$, $\chi_{527}(222,·)$, $\chi_{527}(32,·)$, $\chi_{527}(98,·)$, $\chi_{527}(36,·)$, $\chi_{527}(366,·)$, $\chi_{527}(304,·)$, $\chi_{527}(497,·)$, $\chi_{527}(242,·)$, $\chi_{527}(501,·)$, $\chi_{527}(118,·)$, $\chi_{527}(315,·)$, $\chi_{527}(253,·)$, $\chi_{527}(191,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{20} - \frac{1}{4} a^{19} - \frac{1}{4} a^{18} - \frac{1}{4} a^{17} - \frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{11} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{21} - \frac{1}{4} a^{17} - \frac{1}{4} a^{16} - \frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{2} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{22} - \frac{1}{4} a^{18} - \frac{1}{4} a^{17} - \frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{2} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{382375526843287753703813266458917075803640997326112980409642912143802578254646066012} a^{23} + \frac{5775421512055975691246498621457321283035012552916056568956806728473617526355221839}{382375526843287753703813266458917075803640997326112980409642912143802578254646066012} a^{22} - \frac{7774166058276288935900802159685526970154264387554555829255122935931932273874968047}{191187763421643876851906633229458537901820498663056490204821456071901289127323033006} a^{21} - \frac{11956488641252270346463183168679199621302279701478708541942156515356233959238188477}{382375526843287753703813266458917075803640997326112980409642912143802578254646066012} a^{20} + \frac{45965834467558326464837374827121261939803912901206794343161899439486324495081898655}{191187763421643876851906633229458537901820498663056490204821456071901289127323033006} a^{19} + \frac{94590604486346950303949572724852282563073145845993399530014930239547785205419892025}{382375526843287753703813266458917075803640997326112980409642912143802578254646066012} a^{18} + \frac{15252280266314232915026758210567467525469680179116681051296376509090978609072765197}{191187763421643876851906633229458537901820498663056490204821456071901289127323033006} a^{17} - \frac{67208917159635789813247242176286395936867458780587857330640151225787761251268405655}{382375526843287753703813266458917075803640997326112980409642912143802578254646066012} a^{16} + \frac{35884916248395225688417521094353925424546137029214720241659523365175624748696285105}{191187763421643876851906633229458537901820498663056490204821456071901289127323033006} a^{15} + \frac{729942497105196864114301036902139878582329089130508200871831787544114010428543237}{191187763421643876851906633229458537901820498663056490204821456071901289127323033006} a^{14} - \frac{25563474486753316909874992710654506532324902865489153266248168338602088582527497959}{191187763421643876851906633229458537901820498663056490204821456071901289127323033006} a^{13} + \frac{54853659887394014651344932674596195729064399347422428074456120509002322876967242273}{382375526843287753703813266458917075803640997326112980409642912143802578254646066012} a^{12} - \frac{39576215018103384178133818225450230275877599543989419948337313689317217958372951410}{95593881710821938425953316614729268950910249331528245102410728035950644563661516503} a^{11} - \frac{38167000311356399562738103777771827013405295387525330020542262589269002176803132973}{191187763421643876851906633229458537901820498663056490204821456071901289127323033006} a^{10} - \frac{85326952178767366315116073833035376615114971180412170021671363380429531930232865969}{382375526843287753703813266458917075803640997326112980409642912143802578254646066012} a^{9} + \frac{125916063413938437623234985627896610450240887883654181827986945119249523757231246259}{382375526843287753703813266458917075803640997326112980409642912143802578254646066012} a^{8} + \frac{183889387114756964534377228871995834381062762926393983136646800094214418937820287909}{382375526843287753703813266458917075803640997326112980409642912143802578254646066012} a^{7} - \frac{28213754544203191218409691105554352625589237917422028448009547385254812138025576941}{382375526843287753703813266458917075803640997326112980409642912143802578254646066012} a^{6} - \frac{177676501781099448009404424579671089938698349103491853036463972502831181261342105415}{382375526843287753703813266458917075803640997326112980409642912143802578254646066012} a^{5} - \frac{26564952424232381787942763828296635015807566289083027174265892041611410236517913189}{191187763421643876851906633229458537901820498663056490204821456071901289127323033006} a^{4} + \frac{69424616383616045829380817080296908833154240767306688672071121111953749018975814215}{191187763421643876851906633229458537901820498663056490204821456071901289127323033006} a^{3} + \frac{17795115817859300753543509798313667804800998581082539908451914652479383234763268729}{191187763421643876851906633229458537901820498663056490204821456071901289127323033006} a^{2} - \frac{119254196670268453157825285372322218535160697020601959043286000566263876783900402285}{382375526843287753703813266458917075803640997326112980409642912143802578254646066012} a + \frac{156428090990971354365887699474145834433768971486712207925374918807928237095796315429}{382375526843287753703813266458917075803640997326112980409642912143802578254646066012}$

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:  $23$
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:  \( 161327439996204220 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{24}$ (as 24T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 24
The 24 conjugacy class representatives for $C_{24}$
Character table for $C_{24}$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 3.3.961.1, 4.4.4913.1, 6.6.4537258673.1, \(\Q(\zeta_{17})^+\), 12.12.101142537013451510924177.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{6}$ $24$ $24$ $24$ $24$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{3}$ R $24$ $24$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$

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
17Data not computed
31Data not computed