↑

Properties

Label 24.0.98795254563...7121.1
Degree $24$
Signature $[0, 12]$
Discriminant $3^{12}\cdot 7^{20}\cdot 13^{12}$
Root discriminant $31.61$
Ramified primes $3, 7, 13$
Class number $32$ (GRH)
Class group $[2, 4, 4]$ (GRH)
Galois group $C_2^2\times C_6$ (as 24T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![531441, 177147, -177147, -196830, -65610, 65610, 92583, -51030, -58158, -14283, 16470, 23970, 8719, -7990, 1830, 529, -718, 210, 127, -30, -10, 10, -3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 3*x^22 + 10*x^21 - 10*x^20 - 30*x^19 + 127*x^18 + 210*x^17 - 718*x^16 + 529*x^15 + 1830*x^14 - 7990*x^13 + 8719*x^12 + 23970*x^11 + 16470*x^10 - 14283*x^9 - 58158*x^8 - 51030*x^7 + 92583*x^6 + 65610*x^5 - 65610*x^4 - 196830*x^3 - 177147*x^2 + 177147*x + 531441)
 
gp: K = bnfinit(x^24 - x^23 - 3*x^22 + 10*x^21 - 10*x^20 - 30*x^19 + 127*x^18 + 210*x^17 - 718*x^16 + 529*x^15 + 1830*x^14 - 7990*x^13 + 8719*x^12 + 23970*x^11 + 16470*x^10 - 14283*x^9 - 58158*x^8 - 51030*x^7 + 92583*x^6 + 65610*x^5 - 65610*x^4 - 196830*x^3 - 177147*x^2 + 177147*x + 531441, 1)
 

Normalized defining polynomial

\( x^{24} - x^{23} - 3 x^{22} + 10 x^{21} - 10 x^{20} - 30 x^{19} + 127 x^{18} + 210 x^{17} - 718 x^{16} + 529 x^{15} + 1830 x^{14} - 7990 x^{13} + 8719 x^{12} + 23970 x^{11} + 16470 x^{10} - 14283 x^{9} - 58158 x^{8} - 51030 x^{7} + 92583 x^{6} + 65610 x^{5} - 65610 x^{4} - 196830 x^{3} - 177147 x^{2} + 177147 x + 531441 \)

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:  \(987952545632990645956882874687477121=3^{12}\cdot 7^{20}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.61$
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$$\chi_{273}(64,·)$, $\chi_{273}(1,·)$, $\chi_{273}(194,·)$, $\chi_{273}(131,·)$, $\chi_{273}(142,·)$, $\chi_{273}(79,·)$, $\chi_{273}(272,·)$, $\chi_{273}(209,·)$, $\chi_{273}(25,·)$, $\chi_{273}(155,·)$, $\chi_{273}(92,·)$, $\chi_{273}(157,·)$, $\chi_{273}(38,·)$, $\chi_{273}(103,·)$, $\chi_{273}(40,·)$, $\chi_{273}(220,·)$, $\chi_{273}(170,·)$, $\chi_{273}(235,·)$, $\chi_{273}(116,·)$, $\chi_{273}(53,·)$, $\chi_{273}(118,·)$, $\chi_{273}(233,·)$, $\chi_{273}(248,·)$, $\chi_{273}(181,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{873} a^{14} - \frac{1}{9} a^{13} + \frac{1}{3} a^{12} + \frac{4}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{3} a^{9} + \frac{4}{9} a^{8} + \frac{137}{291} a^{7} - \frac{1}{9} a^{6} - \frac{2}{9} a^{5} - \frac{1}{3} a^{4} - \frac{1}{9} a^{3} - \frac{2}{9} a^{2} - \frac{1}{3} a + \frac{48}{97}$, $\frac{1}{2619} a^{15} - \frac{1}{2619} a^{14} - \frac{1}{9} a^{13} - \frac{5}{27} a^{12} - \frac{4}{27} a^{11} - \frac{1}{9} a^{10} + \frac{4}{27} a^{9} + \frac{331}{873} a^{8} + \frac{74}{2619} a^{7} + \frac{1}{27} a^{6} - \frac{2}{9} a^{5} - \frac{1}{27} a^{4} + \frac{10}{27} a^{3} - \frac{2}{9} a^{2} + \frac{145}{291} a - \frac{16}{97}$, $\frac{1}{7857} a^{16} - \frac{1}{7857} a^{15} - \frac{1}{2619} a^{14} + \frac{4}{81} a^{13} - \frac{4}{81} a^{12} - \frac{4}{27} a^{11} - \frac{14}{81} a^{10} - \frac{542}{2619} a^{9} - \frac{799}{7857} a^{8} - \frac{2009}{7857} a^{7} + \frac{1}{27} a^{6} - \frac{37}{81} a^{5} + \frac{37}{81} a^{4} + \frac{10}{27} a^{3} + \frac{113}{291} a^{2} - \frac{113}{291} a - \frac{16}{97}$, $\frac{1}{23571} a^{17} - \frac{1}{23571} a^{16} - \frac{1}{7857} a^{15} + \frac{10}{23571} a^{14} - \frac{31}{243} a^{13} - \frac{4}{81} a^{12} + \frac{94}{243} a^{11} - \frac{1415}{7857} a^{10} - \frac{8656}{23571} a^{9} - \frac{7247}{23571} a^{8} - \frac{1928}{7857} a^{7} + \frac{98}{243} a^{6} - \frac{17}{243} a^{5} + \frac{37}{81} a^{4} + \frac{16}{873} a^{3} + \frac{275}{873} a^{2} - \frac{113}{291} a + \frac{7}{97}$, $\frac{1}{282852} a^{18} - \frac{1}{70713} a^{17} + \frac{19}{282852} a^{15} - \frac{10}{70713} a^{14} - \frac{599}{2916} a^{12} - \frac{1969}{7857} a^{11} - \frac{12730}{70713} a^{10} - \frac{1265}{2916} a^{9} + \frac{73}{291} a^{8} + \frac{14126}{70713} a^{7} + \frac{337}{2916} a^{6} - \frac{1540}{7857} a^{4} - \frac{2683}{10476} a^{3} + \frac{127}{291} a - \frac{171}{388}$, $\frac{1}{24556362084} a^{19} + \frac{20737}{12278181042} a^{18} + \frac{3778}{2046363507} a^{17} - \frac{211229}{24556362084} a^{16} + \frac{77149}{12278181042} a^{15} + \frac{37780}{2046363507} a^{14} + \frac{19117345}{253158372} a^{13} - \frac{110026013}{4092727014} a^{12} + \frac{856123844}{6139090521} a^{11} - \frac{3649872857}{24556362084} a^{10} - \frac{693609497}{4092727014} a^{9} + \frac{2604919289}{6139090521} a^{8} - \frac{10986764099}{24556362084} a^{7} - \frac{636593}{42193062} a^{6} - \frac{179420500}{682121169} a^{5} + \frac{45534797}{101054988} a^{4} + \frac{5068187}{151582482} a^{3} - \frac{12357083}{25263747} a^{2} + \frac{2508313}{33684996} a - \frac{409913}{5614166}$, $\frac{1}{73669086252} a^{20} - \frac{1}{73669086252} a^{19} - \frac{11735}{12278181042} a^{18} - \frac{65609}{73669086252} a^{17} + \frac{347237}{73669086252} a^{16} - \frac{49337}{12278181042} a^{15} - \frac{656063}{73669086252} a^{14} - \frac{3651484547}{24556362084} a^{13} - \frac{10852182647}{36834543126} a^{12} - \frac{11564369309}{73669086252} a^{11} - \frac{10979535173}{24556362084} a^{10} - \frac{17332551431}{36834543126} a^{9} + \frac{32389359649}{73669086252} a^{8} - \frac{11934537827}{24556362084} a^{7} + \frac{1052974697}{4092727014} a^{6} + \frac{122269745}{2728484676} a^{5} - \frac{132705539}{909494892} a^{4} - \frac{40340921}{151582482} a^{3} - \frac{33479675}{101054988} a^{2} - \frac{14997095}{33684996} a + \frac{1601317}{5614166}$, $\frac{1}{221007258756} a^{21} - \frac{1}{221007258756} a^{20} - \frac{1}{73669086252} a^{19} + \frac{19537}{55251814689} a^{18} + \frac{1273337}{221007258756} a^{17} - \frac{528643}{73669086252} a^{16} - \frac{818237}{55251814689} a^{15} - \frac{19794215}{73669086252} a^{14} - \frac{27284847559}{221007258756} a^{13} - \frac{21598608686}{55251814689} a^{12} + \frac{17889859087}{73669086252} a^{11} + \frac{78964461875}{221007258756} a^{10} - \frac{21563797091}{55251814689} a^{9} + \frac{15311403025}{73669086252} a^{8} - \frac{3542596813}{8185454028} a^{7} + \frac{756298423}{2046363507} a^{6} - \frac{325621601}{909494892} a^{5} - \frac{85196443}{909494892} a^{4} + \frac{1592387}{75791241} a^{3} - \frac{523479}{11228332} a^{2} - \frac{3241199}{33684996} a - \frac{4349529}{11228332}$, $\frac{1}{663021776268} a^{22} - \frac{1}{663021776268} a^{21} - \frac{1}{221007258756} a^{20} + \frac{5}{331510888134} a^{19} + \frac{67463}{663021776268} a^{18} + \frac{3181049}{221007258756} a^{17} - \frac{4906471}{331510888134} a^{16} - \frac{33424445}{221007258756} a^{15} + \frac{167547377}{663021776268} a^{14} + \frac{36513914975}{331510888134} a^{13} - \frac{72483773303}{221007258756} a^{12} + \frac{180484800887}{663021776268} a^{11} - \frac{164244342223}{331510888134} a^{10} + \frac{95336793475}{221007258756} a^{9} + \frac{10509450667}{24556362084} a^{8} - \frac{1101764693}{12278181042} a^{7} - \frac{53335757}{8185454028} a^{6} - \frac{193158361}{909494892} a^{5} - \frac{39665603}{454747446} a^{4} + \frac{112710803}{303164964} a^{3} + \frac{4475901}{11228332} a^{2} - \frac{1063677}{11228332} a + \frac{2249037}{5614166}$, $\frac{1}{1989065328804} a^{23} - \frac{1}{1989065328804} a^{22} - \frac{1}{663021776268} a^{21} + \frac{5}{994532664402} a^{20} - \frac{5}{994532664402} a^{19} + \frac{490369}{331510888134} a^{18} + \frac{12694505}{994532664402} a^{17} - \frac{18212329}{331510888134} a^{16} + \frac{8272333}{994532664402} a^{15} + \frac{235119167}{994532664402} a^{14} - \frac{5878186705}{331510888134} a^{13} + \frac{253027656847}{994532664402} a^{12} - \frac{1820714329}{994532664402} a^{11} - \frac{130959612271}{331510888134} a^{10} + \frac{13319435051}{36834543126} a^{9} - \frac{2825473679}{36834543126} a^{8} + \frac{5668354513}{12278181042} a^{7} - \frac{331785605}{1364242338} a^{6} + \frac{498806995}{1364242338} a^{5} + \frac{74136901}{454747446} a^{4} - \frac{20601347}{50527494} a^{3} + \frac{42618481}{101054988} a^{2} + \frac{13729625}{33684996} a + \frac{2414473}{11228332}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}$, which has order $32$ (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:  \( \frac{1}{101054988} a^{23} - \frac{59541067}{101054988} a^{2} \) (order $42$)
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:  \( 91734319.3933256 \) (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

\(\Q(\sqrt{273}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-91}) \), \(\Q(\sqrt{-39}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{21}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-3}, \sqrt{-91})\), \(\Q(\sqrt{-7}, \sqrt{-39})\), \(\Q(\sqrt{13}, \sqrt{21})\), \(\Q(\sqrt{-3}, \sqrt{13})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{21}, \sqrt{-39})\), \(\Q(\sqrt{-7}, \sqrt{13})\), 6.6.996974433.1, 6.0.64827.1, 6.0.36924979.1, 6.0.142424919.1, \(\Q(\zeta_{7})\), 6.6.5274997.1, \(\Q(\zeta_{21})^+\), 8.0.5554571841.1, 12.0.993958020055671489.2, 12.0.993958020055671489.3, 12.12.993958020055671489.1, 12.0.20284857552156561.1, \(\Q(\zeta_{21})\), 12.0.993958020055671489.1, 12.0.1363454074150441.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{24}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$