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Properties

Label 24.0.97840059750...5625.1
Degree $24$
Signature $[0, 12]$
Discriminant $3^{32}\cdot 5^{18}\cdot 7^{12}$
Root discriminant $38.28$
Ramified primes $3, 5, 7$
Class number $9$ (GRH)
Class group $[3, 3]$ (GRH)
Galois group $C_2\times C_{12}$ (as 24T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7541, -47436, 522033, -1563601, 2561604, -2358960, 600240, 1618335, -2709069, 2131658, -619155, -705375, 1278473, -1193121, 823281, -461214, 218325, -89118, 31555, -9648, 2526, -548, 96, -12, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 12*x^23 + 96*x^22 - 548*x^21 + 2526*x^20 - 9648*x^19 + 31555*x^18 - 89118*x^17 + 218325*x^16 - 461214*x^15 + 823281*x^14 - 1193121*x^13 + 1278473*x^12 - 705375*x^11 - 619155*x^10 + 2131658*x^9 - 2709069*x^8 + 1618335*x^7 + 600240*x^6 - 2358960*x^5 + 2561604*x^4 - 1563601*x^3 + 522033*x^2 - 47436*x + 7541)
 
gp: K = bnfinit(x^24 - 12*x^23 + 96*x^22 - 548*x^21 + 2526*x^20 - 9648*x^19 + 31555*x^18 - 89118*x^17 + 218325*x^16 - 461214*x^15 + 823281*x^14 - 1193121*x^13 + 1278473*x^12 - 705375*x^11 - 619155*x^10 + 2131658*x^9 - 2709069*x^8 + 1618335*x^7 + 600240*x^6 - 2358960*x^5 + 2561604*x^4 - 1563601*x^3 + 522033*x^2 - 47436*x + 7541, 1)
 

Normalized defining polynomial

\( x^{24} - 12 x^{23} + 96 x^{22} - 548 x^{21} + 2526 x^{20} - 9648 x^{19} + 31555 x^{18} - 89118 x^{17} + 218325 x^{16} - 461214 x^{15} + 823281 x^{14} - 1193121 x^{13} + 1278473 x^{12} - 705375 x^{11} - 619155 x^{10} + 2131658 x^{9} - 2709069 x^{8} + 1618335 x^{7} + 600240 x^{6} - 2358960 x^{5} + 2561604 x^{4} - 1563601 x^{3} + 522033 x^{2} - 47436 x + 7541 \)

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:  \(97840059750173910974834598541259765625=3^{32}\cdot 5^{18}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.28$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(315=3^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{315}(64,·)$, $\chi_{315}(1,·)$, $\chi_{315}(202,·)$, $\chi_{315}(139,·)$, $\chi_{315}(76,·)$, $\chi_{315}(13,·)$, $\chi_{315}(274,·)$, $\chi_{315}(211,·)$, $\chi_{315}(148,·)$, $\chi_{315}(22,·)$, $\chi_{315}(286,·)$, $\chi_{315}(223,·)$, $\chi_{315}(97,·)$, $\chi_{315}(34,·)$, $\chi_{315}(232,·)$, $\chi_{315}(169,·)$, $\chi_{315}(106,·)$, $\chi_{315}(43,·)$, $\chi_{315}(307,·)$, $\chi_{315}(244,·)$, $\chi_{315}(181,·)$, $\chi_{315}(118,·)$, $\chi_{315}(253,·)$, $\chi_{315}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{701} a^{18} + \frac{9}{701} a^{17} + \frac{144}{701} a^{16} + \frac{206}{701} a^{15} + \frac{188}{701} a^{14} - \frac{5}{701} a^{13} - \frac{301}{701} a^{12} + \frac{191}{701} a^{11} - \frac{233}{701} a^{10} - \frac{45}{701} a^{9} - \frac{151}{701} a^{8} - \frac{304}{701} a^{7} + \frac{127}{701} a^{6} - \frac{276}{701} a^{5} - \frac{236}{701} a^{4} - \frac{253}{701} a^{3} + \frac{292}{701} a^{2} - \frac{200}{701} a + \frac{134}{701}$, $\frac{1}{701} a^{19} + \frac{63}{701} a^{17} + \frac{312}{701} a^{16} - \frac{264}{701} a^{15} - \frac{295}{701} a^{14} - \frac{256}{701} a^{13} + \frac{96}{701} a^{12} + \frac{151}{701} a^{11} - \frac{51}{701} a^{10} + \frac{254}{701} a^{9} - \frac{347}{701} a^{8} + \frac{59}{701} a^{7} - \frac{17}{701} a^{6} + \frac{145}{701} a^{5} - \frac{232}{701} a^{4} - \frac{235}{701} a^{3} - \frac{24}{701} a^{2} - \frac{169}{701} a + \frac{196}{701}$, $\frac{1}{701} a^{20} - \frac{255}{701} a^{17} - \frac{223}{701} a^{16} + \frac{46}{701} a^{15} - \frac{183}{701} a^{14} - \frac{290}{701} a^{13} + \frac{187}{701} a^{12} - \frac{167}{701} a^{11} + \frac{212}{701} a^{10} - \frac{316}{701} a^{9} - \frac{242}{701} a^{8} + \frac{208}{701} a^{7} - \frac{145}{701} a^{6} + \frac{332}{701} a^{5} - \frac{88}{701} a^{4} - \frac{208}{701} a^{3} - \frac{339}{701} a^{2} + \frac{178}{701} a - \frac{30}{701}$, $\frac{1}{701} a^{21} - \frac{31}{701} a^{17} + \frac{314}{701} a^{16} - \frac{228}{701} a^{15} - \frac{18}{701} a^{14} + \frac{314}{701} a^{13} + \frac{188}{701} a^{12} - \frac{153}{701} a^{11} - \frac{146}{701} a^{10} + \frac{200}{701} a^{9} + \frac{258}{701} a^{8} + \frac{146}{701} a^{7} - \frac{230}{701} a^{6} + \frac{333}{701} a^{5} - \frac{102}{701} a^{4} + \frac{339}{701} a^{3} + \frac{332}{701} a^{2} + \frac{143}{701} a - \frac{179}{701}$, $\frac{1}{701} a^{22} - \frac{108}{701} a^{17} + \frac{30}{701} a^{16} + \frac{59}{701} a^{15} - \frac{167}{701} a^{14} + \frac{33}{701} a^{13} + \frac{330}{701} a^{12} + \frac{167}{701} a^{11} - \frac{13}{701} a^{10} + \frac{265}{701} a^{9} - \frac{329}{701} a^{8} + \frac{160}{701} a^{7} + \frac{64}{701} a^{6} - \frac{246}{701} a^{5} + \frac{33}{701} a^{4} + \frac{200}{701} a^{3} + \frac{82}{701} a^{2} - \frac{70}{701} a - \frac{52}{701}$, $\frac{1}{10049684002016251414229139928321299766173350873753248885799} a^{23} + \frac{4861480010475132606875208694000821245888323311626727388}{10049684002016251414229139928321299766173350873753248885799} a^{22} - \frac{4470953382445184343231443422755175629338247274560066582}{10049684002016251414229139928321299766173350873753248885799} a^{21} - \frac{5881390889800238339762923559128491981778994655610830619}{10049684002016251414229139928321299766173350873753248885799} a^{20} - \frac{4461683649331583271255419580815894593114830117147250956}{10049684002016251414229139928321299766173350873753248885799} a^{19} + \frac{1758600566487775230985759655249234684212198002393364966}{10049684002016251414229139928321299766173350873753248885799} a^{18} + \frac{3335383699137924033427754661325990452904923025332320330855}{10049684002016251414229139928321299766173350873753248885799} a^{17} - \frac{3522517490111628964454829623547449318309596924366681110840}{10049684002016251414229139928321299766173350873753248885799} a^{16} - \frac{5986796146006159335848879391330217405032234582202505757}{14336211129837733829142852964795006799106064013913336499} a^{15} + \frac{2407509601695238519731212860129796709342064497626573554583}{10049684002016251414229139928321299766173350873753248885799} a^{14} + \frac{1929958966623085831097552079654096711996679620316675628590}{10049684002016251414229139928321299766173350873753248885799} a^{13} - \frac{3451112500539386693894567100054086151101694352734204592812}{10049684002016251414229139928321299766173350873753248885799} a^{12} + \frac{4742869196269565624159245468313125908116492871891586519000}{10049684002016251414229139928321299766173350873753248885799} a^{11} + \frac{3017452196353699560459044330953347979854344902943464190091}{10049684002016251414229139928321299766173350873753248885799} a^{10} - \frac{3255394098913438765011808501982087735876682393651779728537}{10049684002016251414229139928321299766173350873753248885799} a^{9} - \frac{3308352521469143657790862613340600206727283007069520584789}{10049684002016251414229139928321299766173350873753248885799} a^{8} - \frac{4501279920550720955007926979228641762473176407847599652450}{10049684002016251414229139928321299766173350873753248885799} a^{7} - \frac{3624981644205000654939208701562439645450076272535518041834}{10049684002016251414229139928321299766173350873753248885799} a^{6} - \frac{1638302237467407779409177982772769070443444411984306302998}{10049684002016251414229139928321299766173350873753248885799} a^{5} - \frac{4290005002926542562024880543726585339746661698652651735182}{10049684002016251414229139928321299766173350873753248885799} a^{4} - \frac{1504777001037801924218125439954893223586303957150299894031}{10049684002016251414229139928321299766173350873753248885799} a^{3} - \frac{1000465985967028579873448314629130466523981465068273001210}{10049684002016251414229139928321299766173350873753248885799} a^{2} - \frac{2413745934031554355444672585649947529277586393561034940532}{10049684002016251414229139928321299766173350873753248885799} a - \frac{3917790064566762935388651605070189908066988643623643752069}{10049684002016251414229139928321299766173350873753248885799}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (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:  \( \frac{556343419484526885979266437503232716041828}{245467397231454706606826155300938692882062984109} a^{23} - \frac{8830712732642022363163513920383347035831822}{245467397231454706606826155300938692882062984109} a^{22} + \frac{79863933297234753807164480893647308908996008}{245467397231454706606826155300938692882062984109} a^{21} - \frac{517487519361648205585942709650876655466776493}{245467397231454706606826155300938692882062984109} a^{20} + \frac{2625919994433133808941690748125084994213520108}{245467397231454706606826155300938692882062984109} a^{19} - \frac{10999696342570952284115875731567087968798701096}{245467397231454706606826155300938692882062984109} a^{18} + \frac{39068069558070662614570957332408343672852309557}{245467397231454706606826155300938692882062984109} a^{17} - \frac{119788451694359259762192129360642501235570117203}{245467397231454706606826155300938692882062984109} a^{16} + \frac{318993665895448936426978255333055404253981832562}{245467397231454706606826155300938692882062984109} a^{15} - \frac{737650440055771685126544005631479209823952505506}{245467397231454706606826155300938692882062984109} a^{14} + \frac{1464681066612302036550188638772478150578054639238}{245467397231454706606826155300938692882062984109} a^{13} - \frac{2434496738534076762962158113649931968582345421761}{245467397231454706606826155300938692882062984109} a^{12} + \frac{3207607925304879052009218037620772953656607257670}{245467397231454706606826155300938692882062984109} a^{11} - \frac{2906106087107961621625135043674761192435272807937}{245467397231454706606826155300938692882062984109} a^{10} + \frac{683140769767755943120799960756504576802157809786}{245467397231454706606826155300938692882062984109} a^{9} + \frac{3191942649932815040606961736289118062358373330188}{245467397231454706606826155300938692882062984109} a^{8} - \frac{6568973325140339727550561775566114653007268100450}{245467397231454706606826155300938692882062984109} a^{7} + \frac{6468975990767744091118596463614849665885822466077}{245467397231454706606826155300938692882062984109} a^{6} - \frac{1930485632930534869663814826483803855868105677235}{245467397231454706606826155300938692882062984109} a^{5} - \frac{4160279599886154484524227317734116510796959816742}{245467397231454706606826155300938692882062984109} a^{4} + \frac{7112625059242367550384959271809022896391267331872}{245467397231454706606826155300938692882062984109} a^{3} - \frac{5647186429574619319948375404735028611270504178617}{245467397231454706606826155300938692882062984109} a^{2} + \frac{2144733696825885446220491358467863887692525016936}{245467397231454706606826155300938692882062984109} a - \frac{219376880452669108312015188655939790192403793953}{245467397231454706606826155300938692882062984109} \) (order $10$)
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:  \( 423268882.10391355 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{12}$ (as 24T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 24
The 24 conjugacy class representatives for $C_2\times C_{12}$
Character table for $C_2\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{5}, \sqrt{-7})\), 4.4.6125.1, \(\Q(\zeta_{5})\), 6.0.281302875.3, 6.6.820125.1, 6.0.2250423.1, 8.0.37515625.1, 12.0.79131307483265625.1, 12.12.9891413435408203125.1, 12.0.84075626953125.1

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 R ${\href{/LocalNumberField/11.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/13.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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
$3$3.12.16.14$x^{12} + 72 x^{11} - 36 x^{10} + 108 x^{9} - 108 x^{8} + 54 x^{7} + 72 x^{6} - 81 x^{5} - 81 x^{4} - 81 x^{3} + 81 x^{2} - 81$$3$$4$$16$$C_{12}$$[2]^{4}$
3.12.16.14$x^{12} + 72 x^{11} - 36 x^{10} + 108 x^{9} - 108 x^{8} + 54 x^{7} + 72 x^{6} - 81 x^{5} - 81 x^{4} - 81 x^{3} + 81 x^{2} - 81$$3$$4$$16$$C_{12}$$[2]^{4}$
5Data not computed
7Data not computed