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Properties

Label 24.0.28565853987...6352.1
Degree $24$
Signature $[0, 12]$
Discriminant $2^{93}\cdot 19^{16}$
Root discriminant $104.47$
Ramified primes $2, 19$
Class number $42121$ (GRH)
Class group $[42121]$ (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![1138840609, -4045865776, 6039046872, -4404732392, 838300088, 1169190712, -925083908, 95850552, 196565146, -91920960, -10729860, 19850880, -3301798, -2216152, 920836, 93936, -118313, 9272, 9424, -2096, -366, 160, 4, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 8*x^23 + 4*x^22 + 160*x^21 - 366*x^20 - 2096*x^19 + 9424*x^18 + 9272*x^17 - 118313*x^16 + 93936*x^15 + 920836*x^14 - 2216152*x^13 - 3301798*x^12 + 19850880*x^11 - 10729860*x^10 - 91920960*x^9 + 196565146*x^8 + 95850552*x^7 - 925083908*x^6 + 1169190712*x^5 + 838300088*x^4 - 4404732392*x^3 + 6039046872*x^2 - 4045865776*x + 1138840609)
 
gp: K = bnfinit(x^24 - 8*x^23 + 4*x^22 + 160*x^21 - 366*x^20 - 2096*x^19 + 9424*x^18 + 9272*x^17 - 118313*x^16 + 93936*x^15 + 920836*x^14 - 2216152*x^13 - 3301798*x^12 + 19850880*x^11 - 10729860*x^10 - 91920960*x^9 + 196565146*x^8 + 95850552*x^7 - 925083908*x^6 + 1169190712*x^5 + 838300088*x^4 - 4404732392*x^3 + 6039046872*x^2 - 4045865776*x + 1138840609, 1)
 

Normalized defining polynomial

\( x^{24} - 8 x^{23} + 4 x^{22} + 160 x^{21} - 366 x^{20} - 2096 x^{19} + 9424 x^{18} + 9272 x^{17} - 118313 x^{16} + 93936 x^{15} + 920836 x^{14} - 2216152 x^{13} - 3301798 x^{12} + 19850880 x^{11} - 10729860 x^{10} - 91920960 x^{9} + 196565146 x^{8} + 95850552 x^{7} - 925083908 x^{6} + 1169190712 x^{5} + 838300088 x^{4} - 4404732392 x^{3} + 6039046872 x^{2} - 4045865776 x + 1138840609 \)

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

Invariants

Degree:  $24$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 12]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2856585398747452538819699104507633428089224036352=2^{93}\cdot 19^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $104.47$
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:  $2, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(608=2^{5}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{608}(1,·)$, $\chi_{608}(83,·)$, $\chi_{608}(387,·)$, $\chi_{608}(577,·)$, $\chi_{608}(457,·)$, $\chi_{608}(11,·)$, $\chi_{608}(467,·)$, $\chi_{608}(273,·)$, $\chi_{608}(163,·)$, $\chi_{608}(121,·)$, $\chi_{608}(267,·)$, $\chi_{608}(153,·)$, $\chi_{608}(539,·)$, $\chi_{608}(315,·)$, $\chi_{608}(353,·)$, $\chi_{608}(419,·)$, $\chi_{608}(49,·)$, $\chi_{608}(425,·)$, $\chi_{608}(235,·)$, $\chi_{608}(305,·)$, $\chi_{608}(115,·)$, $\chi_{608}(201,·)$, $\chi_{608}(505,·)$, $\chi_{608}(571,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{920445997538978261837721103444957309121} a^{22} - \frac{279557130770158442496448828050375949480}{920445997538978261837721103444957309121} a^{21} - \frac{248711855781743237350321221781614195431}{920445997538978261837721103444957309121} a^{20} - \frac{291766615249841137646487180394652645000}{920445997538978261837721103444957309121} a^{19} + \frac{89571593269292740221350900999369190128}{920445997538978261837721103444957309121} a^{18} + \frac{151100600387780592579348260544982764244}{920445997538978261837721103444957309121} a^{17} + \frac{25897215210283783829084711471893896757}{920445997538978261837721103444957309121} a^{16} - \frac{353200381885714261078664272223900381696}{920445997538978261837721103444957309121} a^{15} + \frac{412027861167978884835850761717315100821}{920445997538978261837721103444957309121} a^{14} + \frac{192075284128264334467031344549764412569}{920445997538978261837721103444957309121} a^{13} + \frac{422416943184853603353902430582828480159}{920445997538978261837721103444957309121} a^{12} + \frac{398985685861379335285434517346275466495}{920445997538978261837721103444957309121} a^{11} + \frac{107674390112330677314266075429191862225}{920445997538978261837721103444957309121} a^{10} + \frac{404250755015107920501741369249416518687}{920445997538978261837721103444957309121} a^{9} - \frac{138406523116063088086127191401809005233}{920445997538978261837721103444957309121} a^{8} - \frac{350090532789783380310141441737601949302}{920445997538978261837721103444957309121} a^{7} + \frac{205399521700908337050170596029422021611}{920445997538978261837721103444957309121} a^{6} + \frac{66408984968863443909659360193697261451}{920445997538978261837721103444957309121} a^{5} + \frac{419411106837624071250377491535792487577}{920445997538978261837721103444957309121} a^{4} + \frac{76745056199194256116657108683905417085}{920445997538978261837721103444957309121} a^{3} + \frac{43424172260423861924930038996399949366}{920445997538978261837721103444957309121} a^{2} + \frac{314243747426297722693723371173234390114}{920445997538978261837721103444957309121} a + \frac{447995544257802888758817952231702434691}{920445997538978261837721103444957309121}$, $\frac{1}{336114459345037244610513984526968374982033701786963009} a^{23} + \frac{1687639168967}{336114459345037244610513984526968374982033701786963009} a^{22} + \frac{79640952453945975174950678849577586341912044861682957}{336114459345037244610513984526968374982033701786963009} a^{21} - \frac{89652712147129823107432030022608196366490841266643}{582520726767828846811982642161123700142172793391617} a^{20} - \frac{30749735081547582374514807657698401053895630933373850}{336114459345037244610513984526968374982033701786963009} a^{19} - \frac{143125618472440757803557860085847151852264606366073486}{336114459345037244610513984526968374982033701786963009} a^{18} - \frac{25317686696378682921928510117435274113518709224518069}{336114459345037244610513984526968374982033701786963009} a^{17} - \frac{21722738862237902327020787131608834686700191070334523}{336114459345037244610513984526968374982033701786963009} a^{16} - \frac{43880489046395486064878982736736765507709024001337871}{336114459345037244610513984526968374982033701786963009} a^{15} + \frac{9325574830536348198611522494397819123523333125218788}{336114459345037244610513984526968374982033701786963009} a^{14} - \frac{93022564983163649151021765326131653804241128404956342}{336114459345037244610513984526968374982033701786963009} a^{13} + \frac{27451380947572702991265965084504982655104887137261646}{336114459345037244610513984526968374982033701786963009} a^{12} + \frac{132369102293419914925797602852718412353577747867728247}{336114459345037244610513984526968374982033701786963009} a^{11} + \frac{96705914615124525933967707915266503200851187179189886}{336114459345037244610513984526968374982033701786963009} a^{10} - \frac{65349849272477234119879960564201218270351467510594391}{336114459345037244610513984526968374982033701786963009} a^{9} - \frac{157296657642508542834710655445284131759823100273382318}{336114459345037244610513984526968374982033701786963009} a^{8} + \frac{73457172022849619886164091084537178339443656044041166}{336114459345037244610513984526968374982033701786963009} a^{7} + \frac{81551322142467120919196344583499637790413647715498275}{336114459345037244610513984526968374982033701786963009} a^{6} - \frac{58991207513641245798258310042948185581617388261882330}{336114459345037244610513984526968374982033701786963009} a^{5} + \frac{67937669355163300605776200869516539596212153418903520}{336114459345037244610513984526968374982033701786963009} a^{4} + \frac{5185427437288057562501730405549342647571469088849836}{336114459345037244610513984526968374982033701786963009} a^{3} - \frac{91292465984182004661595141091359510413922874594148612}{336114459345037244610513984526968374982033701786963009} a^{2} - \frac{128472044018647699961928334568199358249063360340565865}{336114459345037244610513984526968374982033701786963009} a + \frac{121913783591843819288138813969034749020680203155783133}{336114459345037244610513984526968374982033701786963009}$

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

Class group and class number

$C_{42121}$, which has order $42121$ (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:  $11$
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:  \( 2829344790.959167 \) (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

\(\Q(\sqrt{2}) \), 3.3.361.1, \(\Q(\zeta_{16})^+\), 6.6.66724352.1, 8.0.2147483648.1, 12.12.145887695661298614272.1

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 R $24$ $24$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{3}$ $24$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ $24$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{2}$ $24$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{8}$ $24$ $24$

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
2Data not computed
19Data not computed