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Properties

Label 24.24.1313089701...6817.1
Degree $24$
Signature $[24, 0]$
Discriminant $13^{20}\cdot 17^{21}$
Root discriminant $101.14$
Ramified primes $13, 17$
Class number $2$ (GRH)
Class group $[2]$ (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![651067, -1549447, -6533360, 15045613, 21375788, -42464033, -41568187, 54635234, 50070137, -35548977, -35220104, 12003549, 14336745, -2098073, -3510649, 159384, 532103, 2233, -49983, -1147, 2801, 63, -84, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 84*x^22 + 63*x^21 + 2801*x^20 - 1147*x^19 - 49983*x^18 + 2233*x^17 + 532103*x^16 + 159384*x^15 - 3510649*x^14 - 2098073*x^13 + 14336745*x^12 + 12003549*x^11 - 35220104*x^10 - 35548977*x^9 + 50070137*x^8 + 54635234*x^7 - 41568187*x^6 - 42464033*x^5 + 21375788*x^4 + 15045613*x^3 - 6533360*x^2 - 1549447*x + 651067)
 
gp: K = bnfinit(x^24 - x^23 - 84*x^22 + 63*x^21 + 2801*x^20 - 1147*x^19 - 49983*x^18 + 2233*x^17 + 532103*x^16 + 159384*x^15 - 3510649*x^14 - 2098073*x^13 + 14336745*x^12 + 12003549*x^11 - 35220104*x^10 - 35548977*x^9 + 50070137*x^8 + 54635234*x^7 - 41568187*x^6 - 42464033*x^5 + 21375788*x^4 + 15045613*x^3 - 6533360*x^2 - 1549447*x + 651067, 1)
 

Normalized defining polynomial

\( x^{24} - x^{23} - 84 x^{22} + 63 x^{21} + 2801 x^{20} - 1147 x^{19} - 49983 x^{18} + 2233 x^{17} + 532103 x^{16} + 159384 x^{15} - 3510649 x^{14} - 2098073 x^{13} + 14336745 x^{12} + 12003549 x^{11} - 35220104 x^{10} - 35548977 x^{9} + 50070137 x^{8} + 54635234 x^{7} - 41568187 x^{6} - 42464033 x^{5} + 21375788 x^{4} + 15045613 x^{3} - 6533360 x^{2} - 1549447 x + 651067 \)

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:  \(1313089701153189172362017790113081686746246646817=13^{20}\cdot 17^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $101.14$
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:  $13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(221=13\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{221}(1,·)$, $\chi_{221}(134,·)$, $\chi_{221}(55,·)$, $\chi_{221}(77,·)$, $\chi_{221}(16,·)$, $\chi_{221}(81,·)$, $\chi_{221}(212,·)$, $\chi_{221}(217,·)$, $\chi_{221}(152,·)$, $\chi_{221}(25,·)$, $\chi_{221}(155,·)$, $\chi_{221}(157,·)$, $\chi_{221}(35,·)$, $\chi_{221}(36,·)$, $\chi_{221}(168,·)$, $\chi_{221}(43,·)$, $\chi_{221}(49,·)$, $\chi_{221}(179,·)$, $\chi_{221}(118,·)$, $\chi_{221}(183,·)$, $\chi_{221}(120,·)$, $\chi_{221}(121,·)$, $\chi_{221}(127,·)$, $\chi_{221}(191,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{47} a^{20} - \frac{22}{47} a^{19} - \frac{6}{47} a^{18} + \frac{11}{47} a^{17} - \frac{7}{47} a^{16} + \frac{10}{47} a^{15} - \frac{13}{47} a^{14} + \frac{12}{47} a^{13} - \frac{12}{47} a^{12} - \frac{7}{47} a^{11} + \frac{15}{47} a^{10} + \frac{18}{47} a^{9} - \frac{5}{47} a^{8} - \frac{13}{47} a^{7} - \frac{6}{47} a^{6} + \frac{18}{47} a^{5} - \frac{13}{47} a^{4} - \frac{4}{47} a^{3} + \frac{11}{47} a^{2} - \frac{16}{47} a - \frac{19}{47}$, $\frac{1}{47} a^{21} - \frac{20}{47} a^{19} + \frac{20}{47} a^{18} - \frac{3}{47} a^{16} + \frac{19}{47} a^{15} + \frac{8}{47} a^{14} + \frac{17}{47} a^{13} + \frac{11}{47} a^{12} + \frac{2}{47} a^{11} + \frac{19}{47} a^{10} + \frac{15}{47} a^{9} + \frac{18}{47} a^{8} - \frac{10}{47} a^{7} - \frac{20}{47} a^{6} + \frac{7}{47} a^{5} - \frac{8}{47} a^{4} + \frac{17}{47} a^{3} - \frac{9}{47} a^{2} + \frac{5}{47} a + \frac{5}{47}$, $\frac{1}{84083} a^{22} + \frac{298}{84083} a^{21} - \frac{223}{84083} a^{20} - \frac{9699}{84083} a^{19} - \frac{31315}{84083} a^{18} - \frac{36640}{84083} a^{17} - \frac{14400}{84083} a^{16} - \frac{7029}{84083} a^{15} - \frac{1117}{84083} a^{14} + \frac{29055}{84083} a^{13} + \frac{41483}{84083} a^{12} + \frac{28168}{84083} a^{11} - \frac{690}{1789} a^{10} - \frac{35309}{84083} a^{9} + \frac{17978}{84083} a^{8} - \frac{18127}{84083} a^{7} + \frac{33006}{84083} a^{6} + \frac{25449}{84083} a^{5} - \frac{14580}{84083} a^{4} - \frac{3390}{84083} a^{3} - \frac{29397}{84083} a^{2} + \frac{1500}{84083} a - \frac{17119}{84083}$, $\frac{1}{398421145246261405961339977156793574087920402713103326367159} a^{23} - \frac{1114319420814205758180197904325148797104511769600834167}{398421145246261405961339977156793574087920402713103326367159} a^{22} + \frac{1202950629711374495076053110991133089501667121258023396380}{398421145246261405961339977156793574087920402713103326367159} a^{21} + \frac{2366078982514405809898644959675998559666483783127623860222}{398421145246261405961339977156793574087920402713103326367159} a^{20} + \frac{17470987786773721122198008073159866107374686602919596681064}{398421145246261405961339977156793574087920402713103326367159} a^{19} + \frac{136277847084250340286006584731226988446315081866168886420673}{398421145246261405961339977156793574087920402713103326367159} a^{18} + \frac{24121981899920111121002671339679196649351259316168757240152}{398421145246261405961339977156793574087920402713103326367159} a^{17} - \frac{121475904693158490223095385052217841679000862810222621715157}{398421145246261405961339977156793574087920402713103326367159} a^{16} + \frac{314268754229832677246850629248962539232172645247662583659}{398421145246261405961339977156793574087920402713103326367159} a^{15} - \frac{74266500714369610717341442703604731397545814092242320552208}{398421145246261405961339977156793574087920402713103326367159} a^{14} + \frac{45600046418493478316301121396000861552799859760382480438013}{398421145246261405961339977156793574087920402713103326367159} a^{13} + \frac{12595194563077805656505536037698986873494767849152067619618}{398421145246261405961339977156793574087920402713103326367159} a^{12} - \frac{90979839843839369318449203115928737892037891033804623079890}{398421145246261405961339977156793574087920402713103326367159} a^{11} - \frac{59523523511785235464356461453878684910201910219360196014298}{398421145246261405961339977156793574087920402713103326367159} a^{10} + \frac{104758036094696313752203595689099808151646348097621209471691}{398421145246261405961339977156793574087920402713103326367159} a^{9} - \frac{28540828557635745348100474137544392423717997505877348715180}{398421145246261405961339977156793574087920402713103326367159} a^{8} - \frac{173053610228765968553114215653009691030357456541379329337821}{398421145246261405961339977156793574087920402713103326367159} a^{7} - \frac{46694087759841551409314582352354911915207043346246605830247}{398421145246261405961339977156793574087920402713103326367159} a^{6} - \frac{105890779586204686478020680998366981423995384340635606321231}{398421145246261405961339977156793574087920402713103326367159} a^{5} + \frac{45524587385255688735403586078051573064466391251745575199888}{398421145246261405961339977156793574087920402713103326367159} a^{4} - \frac{24540287224601302225201908001291752790506117681558389283873}{398421145246261405961339977156793574087920402713103326367159} a^{3} + \frac{21624748784191213998940885361165286090297474475300114014562}{398421145246261405961339977156793574087920402713103326367159} a^{2} + \frac{163613734402694545848516884312200961080809841890115800434297}{398421145246261405961339977156793574087920402713103326367159} a + \frac{53546948587116432044021098432502118447882762557774682686967}{398421145246261405961339977156793574087920402713103326367159}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 20074475366539372 \) (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

\(\Q(\sqrt{17}) \), 3.3.169.1, 4.4.4913.1, 6.6.140320193.1, 8.8.11719682839553.1, 12.12.96735773996756764337.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ $24$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{3}$ $24$ $24$ R R ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ $24$ $24$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{3}$ $24$ $24$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{24}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$

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
$13$13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
17Data not computed