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Properties

Label 24.24.2929835290...3697.1
Degree $24$
Signature $[24, 0]$
Discriminant $3^{12}\cdot 7^{20}\cdot 17^{21}$
Root discriminant $104.58$
Ramified primes $3, 7, 17$
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![-30499937, -224590834, -255401650, 639111352, 1077486700, -525293261, -1370203412, 143637015, 884418690, 21686592, -341602545, -23875060, 84757288, 6683053, -13895634, -983266, 1506639, 83380, -105494, -4015, 4511, 100, -105, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 105*x^22 + 100*x^21 + 4511*x^20 - 4015*x^19 - 105494*x^18 + 83380*x^17 + 1506639*x^16 - 983266*x^15 - 13895634*x^14 + 6683053*x^13 + 84757288*x^12 - 23875060*x^11 - 341602545*x^10 + 21686592*x^9 + 884418690*x^8 + 143637015*x^7 - 1370203412*x^6 - 525293261*x^5 + 1077486700*x^4 + 639111352*x^3 - 255401650*x^2 - 224590834*x - 30499937)
 
gp: K = bnfinit(x^24 - x^23 - 105*x^22 + 100*x^21 + 4511*x^20 - 4015*x^19 - 105494*x^18 + 83380*x^17 + 1506639*x^16 - 983266*x^15 - 13895634*x^14 + 6683053*x^13 + 84757288*x^12 - 23875060*x^11 - 341602545*x^10 + 21686592*x^9 + 884418690*x^8 + 143637015*x^7 - 1370203412*x^6 - 525293261*x^5 + 1077486700*x^4 + 639111352*x^3 - 255401650*x^2 - 224590834*x - 30499937, 1)
 

Normalized defining polynomial

\( x^{24} - x^{23} - 105 x^{22} + 100 x^{21} + 4511 x^{20} - 4015 x^{19} - 105494 x^{18} + 83380 x^{17} + 1506639 x^{16} - 983266 x^{15} - 13895634 x^{14} + 6683053 x^{13} + 84757288 x^{12} - 23875060 x^{11} - 341602545 x^{10} + 21686592 x^{9} + 884418690 x^{8} + 143637015 x^{7} - 1370203412 x^{6} - 525293261 x^{5} + 1077486700 x^{4} + 639111352 x^{3} - 255401650 x^{2} - 224590834 x - 30499937 \)

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:  \(2929835290463322540183276125634099806723880593697=3^{12}\cdot 7^{20}\cdot 17^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $104.58$
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:  $3, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(357=3\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{357}(256,·)$, $\chi_{357}(1,·)$, $\chi_{357}(67,·)$, $\chi_{357}(4,·)$, $\chi_{357}(257,·)$, $\chi_{357}(64,·)$, $\chi_{357}(332,·)$, $\chi_{357}(268,·)$, $\chi_{357}(205,·)$, $\chi_{357}(206,·)$, $\chi_{357}(16,·)$, $\chi_{357}(83,·)$, $\chi_{357}(26,·)$, $\chi_{357}(230,·)$, $\chi_{357}(104,·)$, $\chi_{357}(169,·)$, $\chi_{357}(106,·)$, $\chi_{357}(236,·)$, $\chi_{357}(110,·)$, $\chi_{357}(310,·)$, $\chi_{357}(185,·)$, $\chi_{357}(314,·)$, $\chi_{357}(59,·)$, $\chi_{357}(319,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{13} a^{11} + \frac{4}{13} a^{10} - \frac{2}{13} a^{9} - \frac{2}{13} a^{8} + \frac{2}{13} a^{7} + \frac{5}{13} a^{6} - \frac{3}{13} a^{5} + \frac{2}{13} a^{4} - \frac{3}{13} a^{3} + \frac{4}{13} a^{2} + \frac{5}{13} a$, $\frac{1}{13} a^{12} - \frac{5}{13} a^{10} + \frac{6}{13} a^{9} - \frac{3}{13} a^{8} - \frac{3}{13} a^{7} + \frac{3}{13} a^{6} + \frac{1}{13} a^{5} + \frac{2}{13} a^{4} + \frac{3}{13} a^{3} + \frac{2}{13} a^{2} + \frac{6}{13} a$, $\frac{1}{13} a^{13} - \frac{1}{13} a$, $\frac{1}{13} a^{14} - \frac{1}{13} a^{2}$, $\frac{1}{13} a^{15} - \frac{1}{13} a^{3}$, $\frac{1}{13} a^{16} - \frac{1}{13} a^{4}$, $\frac{1}{169} a^{17} - \frac{4}{169} a^{16} - \frac{3}{169} a^{15} - \frac{4}{169} a^{14} + \frac{3}{169} a^{13} + \frac{2}{169} a^{12} + \frac{6}{169} a^{11} - \frac{51}{169} a^{10} - \frac{6}{13} a^{9} - \frac{44}{169} a^{8} - \frac{20}{169} a^{7} + \frac{36}{169} a^{6} + \frac{48}{169} a^{5} + \frac{7}{169} a^{4} - \frac{35}{169} a^{3} - \frac{20}{169} a^{2} - \frac{2}{13} a$, $\frac{1}{169} a^{18} - \frac{6}{169} a^{16} - \frac{3}{169} a^{15} + \frac{1}{169} a^{13} + \frac{1}{169} a^{12} - \frac{1}{169} a^{11} + \frac{56}{169} a^{10} + \frac{21}{169} a^{9} - \frac{40}{169} a^{8} + \frac{47}{169} a^{7} - \frac{55}{169} a^{6} - \frac{61}{169} a^{5} + \frac{6}{169} a^{4} + \frac{48}{169} a^{3} - \frac{41}{169} a^{2} - \frac{3}{13} a$, $\frac{1}{169} a^{19} - \frac{1}{169} a^{16} - \frac{5}{169} a^{15} + \frac{3}{169} a^{14} + \frac{6}{169} a^{13} - \frac{2}{169} a^{12} + \frac{1}{169} a^{11} - \frac{77}{169} a^{10} - \frac{66}{169} a^{9} + \frac{4}{169} a^{8} + \frac{20}{169} a^{7} - \frac{1}{169} a^{6} + \frac{47}{169} a^{5} + \frac{25}{169} a^{4} - \frac{30}{169} a^{3} - \frac{68}{169} a^{2}$, $\frac{1}{169} a^{20} + \frac{4}{169} a^{16} + \frac{2}{169} a^{14} + \frac{1}{169} a^{13} + \frac{3}{169} a^{12} - \frac{6}{169} a^{11} - \frac{2}{13} a^{10} - \frac{35}{169} a^{9} + \frac{15}{169} a^{8} - \frac{60}{169} a^{7} + \frac{70}{169} a^{6} + \frac{47}{169} a^{5} - \frac{75}{169} a^{4} + \frac{40}{169} a^{3} + \frac{71}{169} a^{2} - \frac{3}{13} a$, $\frac{1}{2197} a^{21} + \frac{5}{2197} a^{20} + \frac{2}{2197} a^{19} - \frac{4}{2197} a^{18} + \frac{4}{2197} a^{17} - \frac{36}{2197} a^{16} - \frac{61}{2197} a^{15} + \frac{82}{2197} a^{14} + \frac{55}{2197} a^{13} + \frac{40}{2197} a^{12} - \frac{37}{2197} a^{11} + \frac{1004}{2197} a^{10} - \frac{337}{2197} a^{9} - \frac{129}{2197} a^{8} - \frac{131}{2197} a^{7} + \frac{459}{2197} a^{6} + \frac{160}{2197} a^{5} - \frac{972}{2197} a^{4} + \frac{669}{2197} a^{3} - \frac{605}{2197} a^{2} + \frac{43}{169} a - \frac{1}{13}$, $\frac{1}{11009167} a^{22} - \frac{415}{11009167} a^{21} + \frac{20470}{11009167} a^{20} + \frac{8893}{11009167} a^{19} - \frac{28866}{11009167} a^{18} + \frac{369}{846859} a^{17} - \frac{85665}{11009167} a^{16} + \frac{139205}{11009167} a^{15} - \frac{8433}{846859} a^{14} - \frac{293954}{11009167} a^{13} - \frac{3876}{11009167} a^{12} + \frac{304585}{11009167} a^{11} + \frac{1473162}{11009167} a^{10} - \frac{5419131}{11009167} a^{9} + \frac{769361}{11009167} a^{8} - \frac{5246038}{11009167} a^{7} + \frac{3869594}{11009167} a^{6} - \frac{79487}{846859} a^{5} + \frac{3627202}{11009167} a^{4} + \frac{141643}{11009167} a^{3} + \frac{1195677}{11009167} a^{2} + \frac{72407}{846859} a + \frac{2032}{65143}$, $\frac{1}{1980874321526601117660534331889355005125795712190941534609} a^{23} + \frac{74513543346658204624526408237648302654758226926892}{1980874321526601117660534331889355005125795712190941534609} a^{22} + \frac{150406337990821896178120108835490521551192871770198173}{1980874321526601117660534331889355005125795712190941534609} a^{21} - \frac{130690287157777316060112497685846765371329503952646916}{1980874321526601117660534331889355005125795712190941534609} a^{20} - \frac{3719778597688031131586645368920377954769941634209924900}{1980874321526601117660534331889355005125795712190941534609} a^{19} - \frac{3910936787916205540356232845871885903417866314028647761}{1980874321526601117660534331889355005125795712190941534609} a^{18} + \frac{254240865575175702092890889921703955912903393762344624}{1980874321526601117660534331889355005125795712190941534609} a^{17} + \frac{17783155710900461182469607216045928905727931666435604601}{1980874321526601117660534331889355005125795712190941534609} a^{16} + \frac{69200328162730143255276546834129765639886752320572082903}{1980874321526601117660534331889355005125795712190941534609} a^{15} - \frac{31078626137567039961568607989790778794280940475879331647}{1980874321526601117660534331889355005125795712190941534609} a^{14} + \frac{57063591527646263478306321573242287946840112688654878500}{1980874321526601117660534331889355005125795712190941534609} a^{13} - \frac{35953775872915314040784647894560034976481706271433794397}{1980874321526601117660534331889355005125795712190941534609} a^{12} - \frac{9711255838509492259159694965265277378387694194784000576}{1980874321526601117660534331889355005125795712190941534609} a^{11} - \frac{753583621066122135023426414102875519123730620202654789123}{1980874321526601117660534331889355005125795712190941534609} a^{10} - \frac{57353085726892684775613319014725253760090317485671213456}{152374947809738547512348794760719615778907362476226271893} a^{9} - \frac{382601525694286621582310718291168632593147140476526280231}{1980874321526601117660534331889355005125795712190941534609} a^{8} + \frac{771080840095319797021187340083335151586282932688227096353}{1980874321526601117660534331889355005125795712190941534609} a^{7} + \frac{319696438048173587958629133734363358703712784494164074967}{1980874321526601117660534331889355005125795712190941534609} a^{6} + \frac{812800371451174154397854129978196131340350503338989405794}{1980874321526601117660534331889355005125795712190941534609} a^{5} + \frac{112192140376096234966971243782498583707986819021206847473}{1980874321526601117660534331889355005125795712190941534609} a^{4} + \frac{701793401365166259811454991106872432018203969145657929537}{1980874321526601117660534331889355005125795712190941534609} a^{3} + \frac{528145565853836682457256772297136396011282960298455731708}{1980874321526601117660534331889355005125795712190941534609} a^{2} + \frac{29328450931903474021518200077088623760772113227686196991}{152374947809738547512348794760719615778907362476226271893} a + \frac{311130567719498110528708959357660503821109905617707245}{11721149831518349808642214981593816598377489421248174761}$

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:  \( 69410144097227096 \) (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

\(\Q(\sqrt{17}) \), \(\Q(\zeta_{7})^+\), 4.4.4913.1, 6.6.11796113.1, 8.8.79803075463713.1, 12.12.683635509017782097.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ R $24$ R $24$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{24}$ R ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ $24$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{3}$ $24$ $24$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$

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
3Data not computed
7Data not computed
17Data not computed