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Properties

Label 24.24.3125701865...0625.1
Degree $24$
Signature $[24, 0]$
Discriminant $3^{32}\cdot 5^{12}\cdot 17^{21}$
Root discriminant $115.42$
Ramified primes $3, 5, 17$
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![-19889, 56223, 2230455, -7054732, -10843575, 45531318, 13298941, -107443881, 7486908, 111946165, -23757957, -56435415, 15539040, 15106521, -4659075, -2279604, 747420, 199791, -67873, -10062, 3477, 270, -93, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 3*x^23 - 93*x^22 + 270*x^21 + 3477*x^20 - 10062*x^19 - 67873*x^18 + 199791*x^17 + 747420*x^16 - 2279604*x^15 - 4659075*x^14 + 15106521*x^13 + 15539040*x^12 - 56435415*x^11 - 23757957*x^10 + 111946165*x^9 + 7486908*x^8 - 107443881*x^7 + 13298941*x^6 + 45531318*x^5 - 10843575*x^4 - 7054732*x^3 + 2230455*x^2 + 56223*x - 19889)
 
gp: K = bnfinit(x^24 - 3*x^23 - 93*x^22 + 270*x^21 + 3477*x^20 - 10062*x^19 - 67873*x^18 + 199791*x^17 + 747420*x^16 - 2279604*x^15 - 4659075*x^14 + 15106521*x^13 + 15539040*x^12 - 56435415*x^11 - 23757957*x^10 + 111946165*x^9 + 7486908*x^8 - 107443881*x^7 + 13298941*x^6 + 45531318*x^5 - 10843575*x^4 - 7054732*x^3 + 2230455*x^2 + 56223*x - 19889, 1)
 

Normalized defining polynomial

\( x^{24} - 3 x^{23} - 93 x^{22} + 270 x^{21} + 3477 x^{20} - 10062 x^{19} - 67873 x^{18} + 199791 x^{17} + 747420 x^{16} - 2279604 x^{15} - 4659075 x^{14} + 15106521 x^{13} + 15539040 x^{12} - 56435415 x^{11} - 23757957 x^{10} + 111946165 x^{9} + 7486908 x^{8} - 107443881 x^{7} + 13298941 x^{6} + 45531318 x^{5} - 10843575 x^{4} - 7054732 x^{3} + 2230455 x^{2} + 56223 x - 19889 \)

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:  \(31257018659087490216031128727244514979613525390625=3^{32}\cdot 5^{12}\cdot 17^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $115.42$
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:  $3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(765=3^{2}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{765}(256,·)$, $\chi_{765}(1,·)$, $\chi_{765}(526,·)$, $\chi_{765}(271,·)$, $\chi_{765}(16,·)$, $\chi_{765}(529,·)$, $\chi_{765}(274,·)$, $\chi_{765}(19,·)$, $\chi_{765}(484,·)$, $\chi_{765}(604,·)$, $\chi_{765}(349,·)$, $\chi_{765}(94,·)$, $\chi_{765}(229,·)$, $\chi_{765}(739,·)$, $\chi_{765}(676,·)$, $\chi_{765}(421,·)$, $\chi_{765}(166,·)$, $\chi_{765}(616,·)$, $\chi_{765}(361,·)$, $\chi_{765}(106,·)$, $\chi_{765}(559,·)$, $\chi_{765}(304,·)$, $\chi_{765}(49,·)$, $\chi_{765}(511,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{13} a^{18} + \frac{1}{13} a^{17} - \frac{6}{13} a^{16} + \frac{2}{13} a^{15} - \frac{2}{13} a^{14} + \frac{1}{13} a^{13} - \frac{1}{13} a^{12} + \frac{1}{13} a^{11} - \frac{2}{13} a^{10} + \frac{1}{13} a^{9} - \frac{1}{13} a^{8} + \frac{3}{13} a^{7} + \frac{2}{13} a^{6} + \frac{2}{13} a^{5} - \frac{4}{13} a^{4} - \frac{1}{13} a^{3} + \frac{4}{13} a^{2} + \frac{6}{13} a + \frac{3}{13}$, $\frac{1}{13} a^{19} + \frac{6}{13} a^{17} - \frac{5}{13} a^{16} - \frac{4}{13} a^{15} + \frac{3}{13} a^{14} - \frac{2}{13} a^{13} + \frac{2}{13} a^{12} - \frac{3}{13} a^{11} + \frac{3}{13} a^{10} - \frac{2}{13} a^{9} + \frac{4}{13} a^{8} - \frac{1}{13} a^{7} - \frac{6}{13} a^{5} + \frac{3}{13} a^{4} + \frac{5}{13} a^{3} + \frac{2}{13} a^{2} - \frac{3}{13} a - \frac{3}{13}$, $\frac{1}{13} a^{20} + \frac{2}{13} a^{17} + \frac{6}{13} a^{16} + \frac{4}{13} a^{15} - \frac{3}{13} a^{14} - \frac{4}{13} a^{13} + \frac{3}{13} a^{12} - \frac{3}{13} a^{11} - \frac{3}{13} a^{10} - \frac{2}{13} a^{9} + \frac{5}{13} a^{8} - \frac{5}{13} a^{7} - \frac{5}{13} a^{6} + \frac{4}{13} a^{5} + \frac{3}{13} a^{4} - \frac{5}{13} a^{3} - \frac{1}{13} a^{2} - \frac{5}{13}$, $\frac{1}{611} a^{21} - \frac{10}{611} a^{20} - \frac{5}{611} a^{19} - \frac{12}{611} a^{18} - \frac{4}{13} a^{17} + \frac{183}{611} a^{16} + \frac{170}{611} a^{15} - \frac{23}{47} a^{14} + \frac{10}{47} a^{13} - \frac{29}{611} a^{12} + \frac{301}{611} a^{11} + \frac{210}{611} a^{10} - \frac{122}{611} a^{9} - \frac{61}{611} a^{8} - \frac{57}{611} a^{7} + \frac{11}{47} a^{6} - \frac{113}{611} a^{5} + \frac{175}{611} a^{4} + \frac{142}{611} a^{3} - \frac{108}{611} a^{2} - \frac{6}{13} a + \frac{49}{611}$, $\frac{1}{611} a^{22} - \frac{11}{611} a^{20} - \frac{15}{611} a^{19} + \frac{21}{611} a^{18} - \frac{287}{611} a^{17} - \frac{256}{611} a^{16} - \frac{197}{611} a^{15} + \frac{7}{611} a^{14} - \frac{92}{611} a^{13} + \frac{58}{611} a^{12} + \frac{71}{611} a^{11} - \frac{43}{611} a^{10} - \frac{12}{611} a^{9} + \frac{21}{47} a^{8} + \frac{43}{611} a^{7} + \frac{283}{611} a^{6} - \frac{203}{611} a^{5} - \frac{223}{611} a^{4} + \frac{137}{611} a^{3} - \frac{46}{611} a^{2} + \frac{284}{611} a + \frac{255}{611}$, $\frac{1}{28782416107463361234466870255570335796299328847649559000376691082609561} a^{23} - \frac{15700840258437619466234815863479018990572377478519890691277953799256}{28782416107463361234466870255570335796299328847649559000376691082609561} a^{22} - \frac{13154113395905890871464407274776459282093192743795112798384164720744}{28782416107463361234466870255570335796299328847649559000376691082609561} a^{21} + \frac{680594981008257339327663488022624189828091404641775222401672620992814}{28782416107463361234466870255570335796299328847649559000376691082609561} a^{20} + \frac{1044334985048116072224226764726085337020973269241874283138964023071209}{28782416107463361234466870255570335796299328847649559000376691082609561} a^{19} - \frac{932131541142778143916346109148389594892032437314986469448549325206924}{28782416107463361234466870255570335796299328847649559000376691082609561} a^{18} + \frac{13569997904739303964679708806662662153084680414337812598407888645007907}{28782416107463361234466870255570335796299328847649559000376691082609561} a^{17} + \frac{7768047745817810429851589725858958893466898995915190872228160132602279}{28782416107463361234466870255570335796299328847649559000376691082609561} a^{16} - \frac{555723352892128825851529203713455113968474381729802688743525200567302}{28782416107463361234466870255570335796299328847649559000376691082609561} a^{15} + \frac{8465449475702019940208006086356841713116478783779222057785452063105122}{28782416107463361234466870255570335796299328847649559000376691082609561} a^{14} + \frac{13022589994212826404722556337525965095279903917070463005779127922812864}{28782416107463361234466870255570335796299328847649559000376691082609561} a^{13} - \frac{13132601407216046968609301918495763800313022484752345986286875308809903}{28782416107463361234466870255570335796299328847649559000376691082609561} a^{12} - \frac{14223814243833998131572417567583472860461236803166920924973749838134127}{28782416107463361234466870255570335796299328847649559000376691082609561} a^{11} - \frac{1661746181193784728113606969172028643889757027232487653697702561988442}{28782416107463361234466870255570335796299328847649559000376691082609561} a^{10} - \frac{12717365156052320072533693430254512886886635823105778169501444642806060}{28782416107463361234466870255570335796299328847649559000376691082609561} a^{9} + \frac{8398208318654902237441962945703278823241127962767972856169665669516419}{28782416107463361234466870255570335796299328847649559000376691082609561} a^{8} + \frac{10324828334550310071380862737954044687258771736567957585720517855547694}{28782416107463361234466870255570335796299328847649559000376691082609561} a^{7} + \frac{8043721945059523177913425142672251257223668464200305924056743170425985}{28782416107463361234466870255570335796299328847649559000376691082609561} a^{6} + \frac{13159422042754772009076587067616546832936694555092545353291736871447368}{28782416107463361234466870255570335796299328847649559000376691082609561} a^{5} - \frac{274464228047934152325436441695696181480425994266961178662705846347909}{28782416107463361234466870255570335796299328847649559000376691082609561} a^{4} - \frac{13346958804246375513135408427492790102008656168576560521514936056246134}{28782416107463361234466870255570335796299328847649559000376691082609561} a^{3} + \frac{5503220077524534916450057739697159901516132166367189988812867707201723}{28782416107463361234466870255570335796299328847649559000376691082609561} a^{2} - \frac{8192452818437284683447517735785524677675828334803659102124864989122639}{28782416107463361234466870255570335796299328847649559000376691082609561} a - \frac{495319335509514347503041355927762501680549868628629211224446188955925}{28782416107463361234466870255570335796299328847649559000376691082609561}$

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:  \( 41783017964045590 \) (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

\(\Q(\sqrt{17}) \), \(\Q(\zeta_{9})^+\), 4.4.4913.1, 6.6.32234193.1, 8.8.256461670625.1, 12.12.5104819233548816337.1

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 R $24$ $24$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{6}$ $24$ $24$ $24$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{3}$ $24$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$

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
3Data not computed
5Data not computed
$17$17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$