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Properties

Label 24.24.1749094578...5472.1
Degree $24$
Signature $[24, 0]$
Discriminant $2^{93}\cdot 3^{12}\cdot 7^{16}$
Root discriminant $92.99$
Ramified primes $2, 3, 7$
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![707041, 8567888, 18033608, -53389816, -100930472, 140684888, 178027020, -186714968, -149189430, 138010544, 67722252, -60737008, -17278534, 16485000, 2372100, -2788784, -130873, 290696, -5984, -17920, 1218, 592, -60, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 8*x^23 - 60*x^22 + 592*x^21 + 1218*x^20 - 17920*x^19 - 5984*x^18 + 290696*x^17 - 130873*x^16 - 2788784*x^15 + 2372100*x^14 + 16485000*x^13 - 17278534*x^12 - 60737008*x^11 + 67722252*x^10 + 138010544*x^9 - 149189430*x^8 - 186714968*x^7 + 178027020*x^6 + 140684888*x^5 - 100930472*x^4 - 53389816*x^3 + 18033608*x^2 + 8567888*x + 707041)
 
gp: K = bnfinit(x^24 - 8*x^23 - 60*x^22 + 592*x^21 + 1218*x^20 - 17920*x^19 - 5984*x^18 + 290696*x^17 - 130873*x^16 - 2788784*x^15 + 2372100*x^14 + 16485000*x^13 - 17278534*x^12 - 60737008*x^11 + 67722252*x^10 + 138010544*x^9 - 149189430*x^8 - 186714968*x^7 + 178027020*x^6 + 140684888*x^5 - 100930472*x^4 - 53389816*x^3 + 18033608*x^2 + 8567888*x + 707041, 1)
 

Normalized defining polynomial

\( x^{24} - 8 x^{23} - 60 x^{22} + 592 x^{21} + 1218 x^{20} - 17920 x^{19} - 5984 x^{18} + 290696 x^{17} - 130873 x^{16} - 2788784 x^{15} + 2372100 x^{14} + 16485000 x^{13} - 17278534 x^{12} - 60737008 x^{11} + 67722252 x^{10} + 138010544 x^{9} - 149189430 x^{8} - 186714968 x^{7} + 178027020 x^{6} + 140684888 x^{5} - 100930472 x^{4} - 53389816 x^{3} + 18033608 x^{2} + 8567888 x + 707041 \)

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:  \(174909457836898599788885373561654160049383145472=2^{93}\cdot 3^{12}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $92.99$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(672=2^{5}\cdot 3\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{672}(611,·)$, $\chi_{672}(1,·)$, $\chi_{672}(515,·)$, $\chi_{672}(193,·)$, $\chi_{672}(457,·)$, $\chi_{672}(11,·)$, $\chi_{672}(529,·)$, $\chi_{672}(275,·)$, $\chi_{672}(659,·)$, $\chi_{672}(121,·)$, $\chi_{672}(25,·)$, $\chi_{672}(155,·)$, $\chi_{672}(107,·)$, $\chi_{672}(289,·)$, $\chi_{672}(347,·)$, $\chi_{672}(337,·)$, $\chi_{672}(169,·)$, $\chi_{672}(323,·)$, $\chi_{672}(491,·)$, $\chi_{672}(625,·)$, $\chi_{672}(179,·)$, $\chi_{672}(361,·)$, $\chi_{672}(505,·)$, $\chi_{672}(443,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{4}{9} a^{5} + \frac{1}{9} a^{4} - \frac{1}{9} a^{3} - \frac{4}{9} a^{2} + \frac{4}{9} a - \frac{2}{9}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} + \frac{4}{9} a^{3} - \frac{1}{3} a^{2} - \frac{1}{9} a + \frac{1}{9}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{4}{9} a^{4} - \frac{1}{3} a^{3} - \frac{1}{9} a^{2} + \frac{1}{9} a$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{1}{9} a^{7} - \frac{4}{9} a^{5} - \frac{4}{9} a^{4} + \frac{2}{9} a^{2} - \frac{4}{9} a + \frac{2}{9}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{11} + \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{2}{9} a^{5} + \frac{1}{9} a^{4} + \frac{1}{9} a^{3} + \frac{1}{9} a^{2} + \frac{1}{3} a + \frac{1}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{11} - \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{1}{9} a^{6} + \frac{2}{9} a^{5} - \frac{1}{3} a^{4} + \frac{2}{9} a^{3} - \frac{2}{9} a^{2} - \frac{1}{3} a - \frac{1}{9}$, $\frac{1}{27} a^{18} + \frac{1}{27} a^{15} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{8} + \frac{1}{9} a^{7} - \frac{4}{27} a^{6} + \frac{1}{9} a^{5} - \frac{1}{9} a^{4} - \frac{1}{27} a^{3} - \frac{2}{9} a^{2} - \frac{1}{9} a - \frac{2}{27}$, $\frac{1}{27} a^{19} + \frac{1}{27} a^{16} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{27} a^{7} + \frac{1}{9} a^{6} + \frac{1}{3} a^{5} - \frac{4}{27} a^{4} - \frac{1}{9} a^{3} + \frac{1}{3} a^{2} + \frac{13}{27} a + \frac{2}{9}$, $\frac{1}{27} a^{20} + \frac{1}{27} a^{17} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{27} a^{8} + \frac{1}{9} a^{7} - \frac{13}{27} a^{5} - \frac{1}{9} a^{4} + \frac{1}{3} a^{3} - \frac{5}{27} a^{2} - \frac{4}{9} a - \frac{1}{3}$, $\frac{1}{3051} a^{21} + \frac{28}{3051} a^{20} - \frac{26}{3051} a^{19} - \frac{10}{1017} a^{18} - \frac{23}{3051} a^{17} - \frac{62}{3051} a^{16} - \frac{133}{3051} a^{15} + \frac{3}{113} a^{14} + \frac{13}{1017} a^{13} + \frac{10}{1017} a^{12} + \frac{31}{339} a^{11} - \frac{8}{113} a^{10} - \frac{487}{3051} a^{9} + \frac{65}{3051} a^{8} + \frac{293}{3051} a^{7} - \frac{151}{1017} a^{6} + \frac{767}{3051} a^{5} - \frac{517}{3051} a^{4} + \frac{1061}{3051} a^{3} + \frac{601}{3051} a^{2} - \frac{1340}{3051} a + \frac{10}{27}$, $\frac{1}{7112226417469654366965887163} a^{22} - \frac{303364913767414029636539}{7112226417469654366965887163} a^{21} - \frac{67315007636231611767835541}{7112226417469654366965887163} a^{20} + \frac{121481826789396982771903150}{7112226417469654366965887163} a^{19} - \frac{8583719202341748105216547}{7112226417469654366965887163} a^{18} + \frac{136837925781599830387127293}{7112226417469654366965887163} a^{17} - \frac{38433913170730243667177806}{2370742139156551455655295721} a^{16} - \frac{358370844006474093891699053}{7112226417469654366965887163} a^{15} - \frac{131022463124348053734955472}{2370742139156551455655295721} a^{14} + \frac{11101160637857437319972027}{2370742139156551455655295721} a^{13} + \frac{2778725320674056332711192}{790247379718850485218431907} a^{12} + \frac{202975100357313250525457843}{2370742139156551455655295721} a^{11} + \frac{585462229787506512114529328}{7112226417469654366965887163} a^{10} - \frac{1058208172866660630446300341}{7112226417469654366965887163} a^{9} - \frac{7783433639711613693633557}{62940056791766852805007851} a^{8} + \frac{659856730900555784016157235}{7112226417469654366965887163} a^{7} - \frac{383327236099153177446954209}{7112226417469654366965887163} a^{6} + \frac{3429435503171552091471338756}{7112226417469654366965887163} a^{5} + \frac{3345310339410461037570664591}{7112226417469654366965887163} a^{4} + \frac{455992145191579164406926818}{2370742139156551455655295721} a^{3} - \frac{2115190782227291396650004573}{7112226417469654366965887163} a^{2} - \frac{899927938435363988095866641}{2370742139156551455655295721} a + \frac{5271824668678863265691369}{62940056791766852805007851}$, $\frac{1}{9566794008707761281617054272831861557} a^{23} + \frac{113518789}{3188931336235920427205684757610620519} a^{22} - \frac{155009722023549833608518351208489}{3188931336235920427205684757610620519} a^{21} - \frac{80508107951362575512456841922102511}{9566794008707761281617054272831861557} a^{20} + \frac{5542646876451937206503626572402533}{1062977112078640142401894919203540173} a^{19} - \frac{16115690985847945898031970082982091}{1062977112078640142401894919203540173} a^{18} - \frac{7827112477187070158225496959643775}{1062977112078640142401894919203540173} a^{17} - \frac{11642347666233933750355300355096917}{354325704026213380800631639734513391} a^{16} + \frac{35907674046535620699350058609839134}{1062977112078640142401894919203540173} a^{15} - \frac{106300571315619973150610925840883124}{3188931336235920427205684757610620519} a^{14} - \frac{65104110748536505187277963216957623}{3188931336235920427205684757610620519} a^{13} - \frac{167223102459090834063537601109831410}{3188931336235920427205684757610620519} a^{12} + \frac{270177835038385122629362338255405182}{9566794008707761281617054272831861557} a^{11} - \frac{110844698213477142294571204000636931}{3188931336235920427205684757610620519} a^{10} + \frac{101908968287462865439989579571257857}{1062977112078640142401894919203540173} a^{9} - \frac{528204204833820406549708533058578508}{9566794008707761281617054272831861557} a^{8} + \frac{40213207543238193290336425226720971}{1062977112078640142401894919203540173} a^{7} + \frac{24384199649500283276804582814984419}{3188931336235920427205684757610620519} a^{6} + \frac{1645559711095154635896700320627646333}{9566794008707761281617054272831861557} a^{5} - \frac{72333712648090881271170295923600188}{3188931336235920427205684757610620519} a^{4} + \frac{281741449334937804959468258017470652}{1062977112078640142401894919203540173} a^{3} - \frac{179977659489505581008812186645771787}{3188931336235920427205684757610620519} a^{2} + \frac{21345283260222268883307097880452040}{3188931336235920427205684757610620519} a + \frac{13342227717962439051000258578708749}{28220631294123189621289245642571863}$

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:  \( 8200108600656127.0 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{16})^+\), 6.6.1229312.1, 8.8.173946175488.1, 12.12.49519263525896192.1

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 R $24$ R $24$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{8}$ $24$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ $24$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ $24$ $24$

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
3Data not computed
$7$7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$