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Properties

Label 24.0.13979723538...7441.1
Degree $24$
Signature $[0, 12]$
Discriminant $3^{12}\cdot 7^{12}\cdot 13^{20}$
Root discriminant $38.85$
Ramified primes $3, 7, 13$
Class number $14$ (GRH)
Class group $[14]$ (GRH)
Galois group $C_2^2\times C_6$ (as 24T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4096, 6144, -27648, 71680, 287488, -248960, -422464, 29088, 472592, 31368, -278496, 26266, 108739, -13113, -31749, 720, 7413, -69, -1193, 37, 149, 3, -16, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 16*x^22 + 3*x^21 + 149*x^20 + 37*x^19 - 1193*x^18 - 69*x^17 + 7413*x^16 + 720*x^15 - 31749*x^14 - 13113*x^13 + 108739*x^12 + 26266*x^11 - 278496*x^10 + 31368*x^9 + 472592*x^8 + 29088*x^7 - 422464*x^6 - 248960*x^5 + 287488*x^4 + 71680*x^3 - 27648*x^2 + 6144*x + 4096)
 
gp: K = bnfinit(x^24 - x^23 - 16*x^22 + 3*x^21 + 149*x^20 + 37*x^19 - 1193*x^18 - 69*x^17 + 7413*x^16 + 720*x^15 - 31749*x^14 - 13113*x^13 + 108739*x^12 + 26266*x^11 - 278496*x^10 + 31368*x^9 + 472592*x^8 + 29088*x^7 - 422464*x^6 - 248960*x^5 + 287488*x^4 + 71680*x^3 - 27648*x^2 + 6144*x + 4096, 1)
 

Normalized defining polynomial

\( x^{24} - x^{23} - 16 x^{22} + 3 x^{21} + 149 x^{20} + 37 x^{19} - 1193 x^{18} - 69 x^{17} + 7413 x^{16} + 720 x^{15} - 31749 x^{14} - 13113 x^{13} + 108739 x^{12} + 26266 x^{11} - 278496 x^{10} + 31368 x^{9} + 472592 x^{8} + 29088 x^{7} - 422464 x^{6} - 248960 x^{5} + 287488 x^{4} + 71680 x^{3} - 27648 x^{2} + 6144 x + 4096 \)

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:  \(139797235388174693456489443830128457441=3^{12}\cdot 7^{12}\cdot 13^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.85$
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:  $3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(273=3\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{273}(64,·)$, $\chi_{273}(1,·)$, $\chi_{273}(134,·)$, $\chi_{273}(139,·)$, $\chi_{273}(272,·)$, $\chi_{273}(209,·)$, $\chi_{273}(146,·)$, $\chi_{273}(211,·)$, $\chi_{273}(22,·)$, $\chi_{273}(218,·)$, $\chi_{273}(155,·)$, $\chi_{273}(92,·)$, $\chi_{273}(29,·)$, $\chi_{273}(160,·)$, $\chi_{273}(230,·)$, $\chi_{273}(43,·)$, $\chi_{273}(113,·)$, $\chi_{273}(244,·)$, $\chi_{273}(181,·)$, $\chi_{273}(118,·)$, $\chi_{273}(55,·)$, $\chi_{273}(251,·)$, $\chi_{273}(62,·)$, $\chi_{273}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{14} + \frac{3}{8} a^{12} - \frac{3}{8} a^{11} - \frac{3}{8} a^{10} - \frac{1}{8} a^{9} + \frac{3}{8} a^{8} - \frac{3}{8} a^{7} + \frac{3}{8} a^{5} - \frac{1}{8} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{80} a^{16} - \frac{3}{80} a^{15} + \frac{1}{40} a^{14} - \frac{1}{16} a^{13} - \frac{9}{80} a^{12} + \frac{27}{80} a^{11} - \frac{11}{80} a^{10} - \frac{11}{80} a^{9} - \frac{5}{16} a^{8} - \frac{9}{40} a^{7} + \frac{19}{80} a^{6} + \frac{9}{80} a^{5} - \frac{11}{80} a^{4} - \frac{1}{4} a^{3} - \frac{7}{20} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{160} a^{17} - \frac{1}{160} a^{16} - \frac{1}{40} a^{15} - \frac{1}{160} a^{14} - \frac{19}{160} a^{13} + \frac{9}{160} a^{12} + \frac{43}{160} a^{11} - \frac{33}{160} a^{10} - \frac{47}{160} a^{9} + \frac{3}{40} a^{8} - \frac{17}{160} a^{7} + \frac{47}{160} a^{6} + \frac{7}{160} a^{5} + \frac{19}{80} a^{4} - \frac{17}{40} a^{3} - \frac{9}{20} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{320} a^{18} - \frac{1}{320} a^{17} - \frac{13}{320} a^{15} - \frac{11}{320} a^{14} - \frac{11}{320} a^{13} - \frac{153}{320} a^{12} - \frac{17}{64} a^{11} + \frac{69}{320} a^{10} - \frac{1}{10} a^{9} + \frac{43}{320} a^{8} - \frac{5}{64} a^{7} + \frac{83}{320} a^{6} + \frac{37}{160} a^{5} + \frac{3}{20} a^{4} + \frac{1}{40} a^{3} - \frac{1}{4} a^{2} - \frac{3}{10} a - \frac{1}{5}$, $\frac{1}{640} a^{19} - \frac{1}{640} a^{18} + \frac{3}{640} a^{16} + \frac{21}{640} a^{15} - \frac{59}{640} a^{14} + \frac{87}{640} a^{13} + \frac{11}{640} a^{12} + \frac{261}{640} a^{11} + \frac{3}{10} a^{10} + \frac{107}{640} a^{9} - \frac{37}{128} a^{8} + \frac{39}{128} a^{7} + \frac{29}{320} a^{6} + \frac{7}{40} a^{5} + \frac{9}{80} a^{4} - \frac{1}{4} a^{3} - \frac{1}{10} a^{2} - \frac{1}{2} a - \frac{2}{5}$, $\frac{1}{6400} a^{20} + \frac{3}{6400} a^{19} - \frac{1}{6400} a^{17} - \frac{3}{1280} a^{16} - \frac{363}{6400} a^{15} + \frac{191}{6400} a^{14} - \frac{29}{256} a^{13} - \frac{7}{1280} a^{12} - \frac{11}{80} a^{11} + \frac{2479}{6400} a^{10} + \frac{1123}{6400} a^{9} + \frac{1307}{6400} a^{8} - \frac{1359}{3200} a^{7} - \frac{219}{1600} a^{6} + \frac{27}{200} a^{5} - \frac{37}{400} a^{4} + \frac{11}{50} a^{3} + \frac{21}{50} a^{2} - \frac{7}{25} a - \frac{7}{25}$, $\frac{1}{12800} a^{21} - \frac{1}{12800} a^{20} + \frac{1}{1600} a^{19} + \frac{19}{12800} a^{18} + \frac{29}{12800} a^{17} - \frac{3}{12800} a^{16} + \frac{263}{12800} a^{15} + \frac{651}{12800} a^{14} - \frac{431}{2560} a^{13} - \frac{3}{16} a^{12} + \frac{3899}{12800} a^{11} + \frac{1247}{12800} a^{10} - \frac{1}{2560} a^{9} - \frac{483}{6400} a^{8} - \frac{239}{800} a^{7} - \frac{29}{100} a^{6} + \frac{287}{800} a^{5} - \frac{23}{100} a^{4} - \frac{13}{100} a^{3} + \frac{27}{100} a^{2} + \frac{21}{50} a + \frac{9}{25}$, $\frac{1}{59758634757145600} a^{22} + \frac{489422961131}{59758634757145600} a^{21} + \frac{1018996242523}{14939658689286400} a^{20} - \frac{41249577734357}{59758634757145600} a^{19} + \frac{27955576350537}{59758634757145600} a^{18} - \frac{42633927541031}{59758634757145600} a^{17} + \frac{251753311897867}{59758634757145600} a^{16} + \frac{2006231179074119}{59758634757145600} a^{15} - \frac{4920195799426527}{59758634757145600} a^{14} + \frac{402258550149337}{2987931737857280} a^{13} - \frac{24763238052498221}{59758634757145600} a^{12} - \frac{13882998721941}{76125649372160} a^{11} + \frac{17801384976218383}{59758634757145600} a^{10} - \frac{2269459209209789}{29879317378572800} a^{9} + \frac{81609607427289}{933728668080400} a^{8} - \frac{570818768191651}{7469829344643200} a^{7} - \frac{1831711739333393}{3734914672321600} a^{6} - \frac{124155345147379}{933728668080400} a^{5} + \frac{230430450994187}{466864334040200} a^{4} + \frac{10107038075913}{116716083510050} a^{3} + \frac{51177401281751}{233432167020100} a^{2} - \frac{13523094509021}{116716083510050} a + \frac{5279071361136}{58358041755025}$, $\frac{1}{119092776744270485578340669586526361600} a^{23} + \frac{489962605301564149719}{119092776744270485578340669586526361600} a^{22} + \frac{112008982118654881311514312583003}{3721649273258452674323145924578948800} a^{21} + \frac{8069961062585462006341751052804651}{119092776744270485578340669586526361600} a^{20} + \frac{68182672594609319017687393754606013}{119092776744270485578340669586526361600} a^{19} + \frac{153539882530487174263702456864136069}{119092776744270485578340669586526361600} a^{18} - \frac{49913339228142260580391561843855237}{23818555348854097115668133917305272320} a^{17} - \frac{614775944896550365288248419212264213}{119092776744270485578340669586526361600} a^{16} + \frac{1035416015118415191142682957351111509}{119092776744270485578340669586526361600} a^{15} - \frac{398818572263485985337243005120668061}{3721649273258452674323145924578948800} a^{14} + \frac{8968943006048630989404965780756790019}{119092776744270485578340669586526361600} a^{13} - \frac{47947326061364641285399774569954519793}{119092776744270485578340669586526361600} a^{12} - \frac{4097949676887847354047590353366538573}{119092776744270485578340669586526361600} a^{11} + \frac{263935134158157816569512477773442849}{11909277674427048557834066958652636160} a^{10} - \frac{3226551234204289709766999804530325803}{14886597093033810697292583698315795200} a^{9} - \frac{398611600837481755726959115791106981}{2977319418606762139458516739663159040} a^{8} + \frac{57559497640165887597378235784256373}{186082463662922633716157296228947440} a^{7} + \frac{130445146789849020021355794750101909}{744329854651690534864629184915789760} a^{6} + \frac{44506470167562617053767271890420537}{465206159157306584290393240572368600} a^{5} - \frac{431983849251028724037618686258024847}{930412318314613168580786481144737200} a^{4} + \frac{25810102568740182804329232609394353}{58150769894663323036299155071546075} a^{3} + \frac{1551716423608338506395640858576743}{232603079578653292145196620286184300} a^{2} - \frac{27085737437295542487711913095325109}{58150769894663323036299155071546075} a + \frac{22110522317637738720722230885852028}{58150769894663323036299155071546075}$

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

Class group and class number

$C_{14}$, which has order $14$ (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:  \( -\frac{207874974968377265534886033}{2190693918619310524521103283200} a^{23} + \frac{167608973605034312026738339}{1095346959309655262260551641600} a^{22} + \frac{3183051578919232015283958967}{2190693918619310524521103283200} a^{21} - \frac{524176514566806739523053831}{438138783723862104904220656640} a^{20} - \frac{607213327493149588327661001}{43813878372386210490422065664} a^{19} + \frac{1357090145038689228557646141}{273836739827413815565137910400} a^{18} + \frac{125176860764040579738729762921}{1095346959309655262260551641600} a^{17} - \frac{8423459516653161363869960721}{136918369913706907782568955200} a^{16} - \frac{764745633312903284205992653703}{1095346959309655262260551641600} a^{15} + \frac{768897502892836052123091777169}{2190693918619310524521103283200} a^{14} + \frac{6569307593907132624892841912853}{2190693918619310524521103283200} a^{13} - \frac{291724752366557406165107240089}{547673479654827631130275820800} a^{12} - \frac{5932052789597109499967988671079}{547673479654827631130275820800} a^{11} + \frac{7938925101848231402890274076557}{2190693918619310524521103283200} a^{10} + \frac{29552644677020112902897963354093}{1095346959309655262260551641600} a^{9} - \frac{10070328483141517710344642960851}{547673479654827631130275820800} a^{8} - \frac{2783952040669454041991380603867}{68459184956853453891284477600} a^{7} + \frac{757601679668844490869366557603}{34229592478426726945642238800} a^{6} + \frac{658060463215939011065006281437}{17114796239213363472821119400} a^{5} + \frac{86699588936858184831953876793}{34229592478426726945642238800} a^{4} - \frac{642525541282004911087203327631}{17114796239213363472821119400} a^{3} + \frac{84983576815621720539857131921}{8557398119606681736410559700} a^{2} + \frac{2078700045498864500883984031}{855739811960668173641055970} a - \frac{2192097276609451198411577472}{2139349529901670434102639925} \) (order $6$)
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:  \( 327649563.99093217 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_6$ (as 24T3):

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 $C_2^2\times C_6$
Character table for $C_2^2\times C_6$ is not computed

Intermediate fields

\(\Q(\sqrt{273}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-91}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{13}) \), 3.3.169.1, \(\Q(\sqrt{-3}, \sqrt{-91})\), \(\Q(\sqrt{-7}, \sqrt{-39})\), \(\Q(\sqrt{13}, \sqrt{21})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{13})\), \(\Q(\sqrt{-7}, \sqrt{13})\), \(\Q(\sqrt{21}, \sqrt{-39})\), 6.6.3438544473.1, 6.0.771147.1, 6.0.127353499.1, 6.0.9796423.1, 6.0.10024911.1, 6.6.264503421.1, \(\Q(\zeta_{13})^+\), 8.0.5554571841.1, 12.0.11823588092798847729.3, 12.0.11823588092798847729.1, 12.12.11823588092798847729.1, 12.0.69962059720703241.1, 12.0.100498840557921.1, 12.0.16218913707543001.1, 12.0.11823588092798847729.2

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{12}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.12.6.1$x^{12} + 294 x^{8} + 3430 x^{6} + 21609 x^{4} + 487403 x^{2} + 2941225$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
7.12.6.1$x^{12} + 294 x^{8} + 3430 x^{6} + 21609 x^{4} + 487403 x^{2} + 2941225$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$13$13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$