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Properties

Label 24.24.5894158243...8841.1
Degree $24$
Signature $[24, 0]$
Discriminant $7^{20}\cdot 41^{21}$
Root discriminant $130.45$
Ramified primes $7, 41$
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![-652259, -1293067, 16409119, 32025510, -99840540, -255245949, 35041069, 432110280, 162034932, -282394963, -174618685, 79995192, 68958618, -10841299, -14117143, 635675, 1676454, 536, -119135, -1821, 4952, 77, -110, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 110*x^22 + 77*x^21 + 4952*x^20 - 1821*x^19 - 119135*x^18 + 536*x^17 + 1676454*x^16 + 635675*x^15 - 14117143*x^14 - 10841299*x^13 + 68958618*x^12 + 79995192*x^11 - 174618685*x^10 - 282394963*x^9 + 162034932*x^8 + 432110280*x^7 + 35041069*x^6 - 255245949*x^5 - 99840540*x^4 + 32025510*x^3 + 16409119*x^2 - 1293067*x - 652259)
 
gp: K = bnfinit(x^24 - x^23 - 110*x^22 + 77*x^21 + 4952*x^20 - 1821*x^19 - 119135*x^18 + 536*x^17 + 1676454*x^16 + 635675*x^15 - 14117143*x^14 - 10841299*x^13 + 68958618*x^12 + 79995192*x^11 - 174618685*x^10 - 282394963*x^9 + 162034932*x^8 + 432110280*x^7 + 35041069*x^6 - 255245949*x^5 - 99840540*x^4 + 32025510*x^3 + 16409119*x^2 - 1293067*x - 652259, 1)
 

Normalized defining polynomial

\( x^{24} - x^{23} - 110 x^{22} + 77 x^{21} + 4952 x^{20} - 1821 x^{19} - 119135 x^{18} + 536 x^{17} + 1676454 x^{16} + 635675 x^{15} - 14117143 x^{14} - 10841299 x^{13} + 68958618 x^{12} + 79995192 x^{11} - 174618685 x^{10} - 282394963 x^{9} + 162034932 x^{8} + 432110280 x^{7} + 35041069 x^{6} - 255245949 x^{5} - 99840540 x^{4} + 32025510 x^{3} + 16409119 x^{2} - 1293067 x - 652259 \)

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:  \(589415824352273084266952490343550409844469452348841=7^{20}\cdot 41^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $130.45$
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:  $7, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(287=7\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{287}(1,·)$, $\chi_{287}(3,·)$, $\chi_{287}(68,·)$, $\chi_{287}(214,·)$, $\chi_{287}(9,·)$, $\chi_{287}(202,·)$, $\chi_{287}(247,·)$, $\chi_{287}(204,·)$, $\chi_{287}(208,·)$, $\chi_{287}(81,·)$, $\chi_{287}(150,·)$, $\chi_{287}(27,·)$, $\chi_{287}(32,·)$, $\chi_{287}(163,·)$, $\chi_{287}(165,·)$, $\chi_{287}(38,·)$, $\chi_{287}(167,·)$, $\chi_{287}(178,·)$, $\chi_{287}(96,·)$, $\chi_{287}(50,·)$, $\chi_{287}(243,·)$, $\chi_{287}(155,·)$, $\chi_{287}(55,·)$, $\chi_{287}(114,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{57286} a^{22} - \frac{14083}{57286} a^{21} + \frac{11853}{57286} a^{20} - \frac{7883}{57286} a^{19} - \frac{2819}{57286} a^{18} + \frac{6639}{57286} a^{17} - \frac{7743}{28643} a^{16} + \frac{12047}{57286} a^{15} + \frac{1545}{57286} a^{14} - \frac{8600}{28643} a^{13} + \frac{21833}{57286} a^{12} + \frac{17123}{57286} a^{11} - \frac{12165}{28643} a^{10} - \frac{14099}{57286} a^{9} + \frac{20713}{57286} a^{8} + \frac{12947}{28643} a^{7} - \frac{20587}{57286} a^{6} + \frac{5045}{57286} a^{5} + \frac{5539}{57286} a^{4} + \frac{13946}{28643} a^{3} + \frac{9327}{28643} a^{2} - \frac{19363}{57286} a + \frac{17369}{57286}$, $\frac{1}{282567859156635280124081170364359957403496919484287021572085141968518157230482462} a^{23} - \frac{722157113129712023425224419298161643332208101807327450647440237985466347265}{141283929578317640062040585182179978701748459742143510786042570984259078615241231} a^{22} + \frac{4533772564983122391622313228408818866019767088473985690665403522881536690165392}{141283929578317640062040585182179978701748459742143510786042570984259078615241231} a^{21} + \frac{22600546697070150646928788598931533825800039605069479795792559851054063504729756}{141283929578317640062040585182179978701748459742143510786042570984259078615241231} a^{20} + \frac{9528645856772289229235351936726326826245614960101310361177136975770755382501745}{282567859156635280124081170364359957403496919484287021572085141968518157230482462} a^{19} + \frac{23134266964424482333720461338585767843758522721523014774259656589087819123946300}{141283929578317640062040585182179978701748459742143510786042570984259078615241231} a^{18} + \frac{45442604913988349186907029831157681312356796325466893485748896376862612702554045}{141283929578317640062040585182179978701748459742143510786042570984259078615241231} a^{17} + \frac{85627112108690337327712044790700985170812119246789868548132636224077416718885499}{282567859156635280124081170364359957403496919484287021572085141968518157230482462} a^{16} - \frac{8715218872061121340253817093544361881694028576538259854564042164426341821103819}{141283929578317640062040585182179978701748459742143510786042570984259078615241231} a^{15} - \frac{10354640367487979166367076604225805323669942484754467547015963886533018528807130}{141283929578317640062040585182179978701748459742143510786042570984259078615241231} a^{14} - \frac{88960777140244709090658796806218321710415814263027645165146897164509733901258581}{282567859156635280124081170364359957403496919484287021572085141968518157230482462} a^{13} + \frac{76453094083474271462845509185729888212640726814527669686300565443586238692993311}{282567859156635280124081170364359957403496919484287021572085141968518157230482462} a^{12} - \frac{121497098072588629207616039822752974699322061987198330918932437071557012367537711}{282567859156635280124081170364359957403496919484287021572085141968518157230482462} a^{11} + \frac{1953872227043598590389028931243842220784544258698606477803920683667821401430852}{141283929578317640062040585182179978701748459742143510786042570984259078615241231} a^{10} + \frac{58690310125237427082990119419272042829943914746495276499415507846138017268104324}{141283929578317640062040585182179978701748459742143510786042570984259078615241231} a^{9} + \frac{3784429171132420981452741399914376229193995386487218512074916349326408855617293}{282567859156635280124081170364359957403496919484287021572085141968518157230482462} a^{8} - \frac{92308637278340405234065562596182824188483535006111940800909860584834461743522201}{282567859156635280124081170364359957403496919484287021572085141968518157230482462} a^{7} + \frac{50981970375237161909967327328580414457308112497926915580308379092493065090947329}{141283929578317640062040585182179978701748459742143510786042570984259078615241231} a^{6} + \frac{135457746052030975713359349393711680052361744820214095087243146195346130958811371}{282567859156635280124081170364359957403496919484287021572085141968518157230482462} a^{5} - \frac{41053334694555372550538536669549280483898690632092285783024875394452159222756230}{141283929578317640062040585182179978701748459742143510786042570984259078615241231} a^{4} + \frac{2937721412449816701180404201560739998598342510419126953610500971305838875455255}{141283929578317640062040585182179978701748459742143510786042570984259078615241231} a^{3} + \frac{56893179319233604805744358478648149319333489194477738960580025122445223964574159}{141283929578317640062040585182179978701748459742143510786042570984259078615241231} a^{2} - \frac{120627178043612121156337234045801114945169849645555564576691949863385360374787941}{282567859156635280124081170364359957403496919484287021572085141968518157230482462} a + \frac{354138547067243576315598323713227432469212296130048253766614922659081658348331}{745561633658668285287813114417836299217669972254055465889406707040945005885178}$

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:  \( 310155799064182340 \) (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

\(\Q(\sqrt{41}) \), \(\Q(\zeta_{7})^+\), 4.4.68921.1, 6.6.165479321.1, 8.8.467605011588281.1, 12.12.1887291702776240766761.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ $24$ ${\href{/LocalNumberField/5.12.0.1}{12} }^{2}$ R $24$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{3}$ $24$ $24$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{8}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{6}$ $24$ $24$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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
7Data not computed
$41$41.8.7.1$x^{8} - 41$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.1$x^{8} - 41$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.1$x^{8} - 41$$8$$1$$7$$C_8$$[\ ]_{8}$