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Properties

Label 24.24.4597492038...2497.1
Degree $24$
Signature $[24, 0]$
Discriminant $13^{16}\cdot 17^{21}$
Root discriminant $65.96$
Ramified primes $13, 17$
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![-307, 417, 37133, -93592, -236349, 737208, 378789, -1994333, 1574, 2610189, -478385, -1920029, 474876, 853425, -220835, -235712, 58160, 40211, -9167, -4070, 863, 222, -45, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 5*x^23 - 45*x^22 + 222*x^21 + 863*x^20 - 4070*x^19 - 9167*x^18 + 40211*x^17 + 58160*x^16 - 235712*x^15 - 220835*x^14 + 853425*x^13 + 474876*x^12 - 1920029*x^11 - 478385*x^10 + 2610189*x^9 + 1574*x^8 - 1994333*x^7 + 378789*x^6 + 737208*x^5 - 236349*x^4 - 93592*x^3 + 37133*x^2 + 417*x - 307)
 
gp: K = bnfinit(x^24 - 5*x^23 - 45*x^22 + 222*x^21 + 863*x^20 - 4070*x^19 - 9167*x^18 + 40211*x^17 + 58160*x^16 - 235712*x^15 - 220835*x^14 + 853425*x^13 + 474876*x^12 - 1920029*x^11 - 478385*x^10 + 2610189*x^9 + 1574*x^8 - 1994333*x^7 + 378789*x^6 + 737208*x^5 - 236349*x^4 - 93592*x^3 + 37133*x^2 + 417*x - 307, 1)
 

Normalized defining polynomial

\( x^{24} - 5 x^{23} - 45 x^{22} + 222 x^{21} + 863 x^{20} - 4070 x^{19} - 9167 x^{18} + 40211 x^{17} + 58160 x^{16} - 235712 x^{15} - 220835 x^{14} + 853425 x^{13} + 474876 x^{12} - 1920029 x^{11} - 478385 x^{10} + 2610189 x^{9} + 1574 x^{8} - 1994333 x^{7} + 378789 x^{6} + 737208 x^{5} - 236349 x^{4} - 93592 x^{3} + 37133 x^{2} + 417 x - 307 \)

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:  \(45974920386302621489514295371768554558532497=13^{16}\cdot 17^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.96$
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$$\chi_{221}(1,·)$, $\chi_{221}(66,·)$, $\chi_{221}(196,·)$, $\chi_{221}(9,·)$, $\chi_{221}(183,·)$, $\chi_{221}(16,·)$, $\chi_{221}(81,·)$, $\chi_{221}(87,·)$, $\chi_{221}(152,·)$, $\chi_{221}(217,·)$, $\chi_{221}(157,·)$, $\chi_{221}(94,·)$, $\chi_{221}(144,·)$, $\chi_{221}(35,·)$, $\chi_{221}(100,·)$, $\chi_{221}(42,·)$, $\chi_{221}(172,·)$, $\chi_{221}(178,·)$, $\chi_{221}(53,·)$, $\chi_{221}(118,·)$, $\chi_{221}(55,·)$, $\chi_{221}(120,·)$, $\chi_{221}(185,·)$, $\chi_{221}(191,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{103} a^{22} - \frac{26}{103} a^{21} - \frac{47}{103} a^{20} + \frac{7}{103} a^{19} + \frac{1}{103} a^{18} + \frac{4}{103} a^{17} - \frac{14}{103} a^{16} - \frac{3}{103} a^{15} - \frac{25}{103} a^{14} - \frac{42}{103} a^{13} - \frac{47}{103} a^{12} - \frac{29}{103} a^{11} + \frac{43}{103} a^{10} + \frac{48}{103} a^{9} - \frac{8}{103} a^{8} - \frac{11}{103} a^{7} + \frac{9}{103} a^{6} + \frac{24}{103} a^{5} - \frac{22}{103} a^{4} + \frac{16}{103} a^{3} + \frac{14}{103} a^{2} + \frac{37}{103} a + \frac{50}{103}$, $\frac{1}{25886899426964038960657910078693150799084693064272557} a^{23} + \frac{75810506467272955447271510863312815768149824038349}{25886899426964038960657910078693150799084693064272557} a^{22} - \frac{3941686623983198338640999427207790475732083863113401}{25886899426964038960657910078693150799084693064272557} a^{21} - \frac{11472792806845960677347013092166489144505707344637839}{25886899426964038960657910078693150799084693064272557} a^{20} - \frac{8057722908081837346222587159112404476712589805610565}{25886899426964038960657910078693150799084693064272557} a^{19} - \frac{5526287018916897567461107351280282103785584823944138}{25886899426964038960657910078693150799084693064272557} a^{18} - \frac{3036696800095890006724565774507443482950328788991327}{25886899426964038960657910078693150799084693064272557} a^{17} + \frac{6664262588674702664322364354554604887176080345255194}{25886899426964038960657910078693150799084693064272557} a^{16} - \frac{472361254982043616787164425079082093613065554304690}{25886899426964038960657910078693150799084693064272557} a^{15} + \frac{606176635898567787030265158501538397483749340690307}{25886899426964038960657910078693150799084693064272557} a^{14} - \frac{3948018926696172614951829428724114465698541871240457}{25886899426964038960657910078693150799084693064272557} a^{13} - \frac{10046310849663443959471424217573115127562942124204970}{25886899426964038960657910078693150799084693064272557} a^{12} + \frac{4731170423456889309582891924077708262487743783144922}{25886899426964038960657910078693150799084693064272557} a^{11} + \frac{12374660069239606226369929828966724289884698372022479}{25886899426964038960657910078693150799084693064272557} a^{10} - \frac{9922865665240451614964071828025719563079009009898993}{25886899426964038960657910078693150799084693064272557} a^{9} + \frac{1212705653167904336921584392185243413431202656108436}{25886899426964038960657910078693150799084693064272557} a^{8} + \frac{990243214944977826658047167890137308184642851465969}{25886899426964038960657910078693150799084693064272557} a^{7} - \frac{12597786091582285764158953352099555035223462667071934}{25886899426964038960657910078693150799084693064272557} a^{6} - \frac{7532029702756924925668979019765344974737428399388359}{25886899426964038960657910078693150799084693064272557} a^{5} - \frac{3658053890759377926002529232764610176963065531059192}{25886899426964038960657910078693150799084693064272557} a^{4} + \frac{1595698899785817765650773978568128682079702442568793}{25886899426964038960657910078693150799084693064272557} a^{3} + \frac{2755878498813868341553118275954591793766708682057640}{25886899426964038960657910078693150799084693064272557} a^{2} - \frac{11560691640307718794981544285402236946541580341128380}{25886899426964038960657910078693150799084693064272557} a + \frac{4594015336012596599553887449116333545000881207424469}{25886899426964038960657910078693150799084693064272557}$

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:  \( 116275529261718.16 \) (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, \(\Q(\zeta_{17})^+\), 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 ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ $24$ $24$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{3}$ $24$ $24$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$

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$13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
17Data not computed