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Properties

Label 21.5.37877966044...6336.1
Degree $21$
Signature $[5, 8]$
Discriminant $2^{24}\cdot 11^{18}\cdot 67^{8}$
Root discriminant $85.57$
Ramified primes $2, 11, 67$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $\PSL(3,4)$ (as 21T67)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-22322, -103930, 463496, -539862, 514117, -1307021, 2097737, -1475463, 190336, 356326, -179184, -29952, 36114, 6884, -10782, 1538, 1844, -1128, 247, 1, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 9*x^20 + x^19 + 247*x^18 - 1128*x^17 + 1844*x^16 + 1538*x^15 - 10782*x^14 + 6884*x^13 + 36114*x^12 - 29952*x^11 - 179184*x^10 + 356326*x^9 + 190336*x^8 - 1475463*x^7 + 2097737*x^6 - 1307021*x^5 + 514117*x^4 - 539862*x^3 + 463496*x^2 - 103930*x - 22322)
 
gp: K = bnfinit(x^21 - 9*x^20 + x^19 + 247*x^18 - 1128*x^17 + 1844*x^16 + 1538*x^15 - 10782*x^14 + 6884*x^13 + 36114*x^12 - 29952*x^11 - 179184*x^10 + 356326*x^9 + 190336*x^8 - 1475463*x^7 + 2097737*x^6 - 1307021*x^5 + 514117*x^4 - 539862*x^3 + 463496*x^2 - 103930*x - 22322, 1)
 

Normalized defining polynomial

\( x^{21} - 9 x^{20} + x^{19} + 247 x^{18} - 1128 x^{17} + 1844 x^{16} + 1538 x^{15} - 10782 x^{14} + 6884 x^{13} + 36114 x^{12} - 29952 x^{11} - 179184 x^{10} + 356326 x^{9} + 190336 x^{8} - 1475463 x^{7} + 2097737 x^{6} - 1307021 x^{5} + 514117 x^{4} - 539862 x^{3} + 463496 x^{2} - 103930 x - 22322 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(37877966044500313823897559260772576526336=2^{24}\cdot 11^{18}\cdot 67^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $85.57$
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:  $2, 11, 67$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{11} a^{18} - \frac{2}{11} a^{17} - \frac{3}{11} a^{15} + \frac{3}{11} a^{14} + \frac{5}{11} a^{12} - \frac{1}{11} a^{11} + \frac{1}{11} a^{10} - \frac{3}{11} a^{9} - \frac{3}{11} a^{8} - \frac{3}{11} a^{7} - \frac{1}{11} a^{6} + \frac{2}{11} a^{5} + \frac{2}{11} a^{4} + \frac{4}{11} a^{3} + \frac{1}{11} a^{2} - \frac{5}{11} a + \frac{2}{11}$, $\frac{1}{11} a^{19} - \frac{4}{11} a^{17} - \frac{3}{11} a^{16} - \frac{3}{11} a^{15} - \frac{5}{11} a^{14} + \frac{5}{11} a^{13} - \frac{2}{11} a^{12} - \frac{1}{11} a^{11} - \frac{1}{11} a^{10} + \frac{2}{11} a^{9} + \frac{2}{11} a^{8} + \frac{4}{11} a^{7} - \frac{5}{11} a^{5} - \frac{3}{11} a^{4} - \frac{2}{11} a^{3} - \frac{3}{11} a^{2} + \frac{3}{11} a + \frac{4}{11}$, $\frac{1}{117615855312388741966883503875902045280188309477291452895538742} a^{20} + \frac{181714291946261071473969908601621353159180848251384130031121}{58807927656194370983441751937951022640094154738645726447769371} a^{19} + \frac{3649942228676169070653513375262379000186000452835484291513723}{117615855312388741966883503875902045280188309477291452895538742} a^{18} + \frac{20822203646409466197461319860913093070125450001609223294479200}{58807927656194370983441751937951022640094154738645726447769371} a^{17} - \frac{16923541662534249416985463621813414296318804134951219937626177}{58807927656194370983441751937951022640094154738645726447769371} a^{16} - \frac{17652362291191499691546680350081694177343575838715741250911262}{58807927656194370983441751937951022640094154738645726447769371} a^{15} - \frac{27386096196716435252096098632390332428969918068342070881486134}{58807927656194370983441751937951022640094154738645726447769371} a^{14} - \frac{8845012454704026092209653334602809645869761306037808127144357}{58807927656194370983441751937951022640094154738645726447769371} a^{13} - \frac{11918060360783930114378343095897530787946576392085804030329468}{58807927656194370983441751937951022640094154738645726447769371} a^{12} + \frac{13582125729719174575036440113336604688568157669073593281025655}{58807927656194370983441751937951022640094154738645726447769371} a^{11} + \frac{637484580344865573109567396586948959495245935744204068864372}{58807927656194370983441751937951022640094154738645726447769371} a^{10} + \frac{6054004576215063085841353665734201169397325981630207308819541}{58807927656194370983441751937951022640094154738645726447769371} a^{9} + \frac{25953461653030591120237385553251022715073825008666887000256774}{58807927656194370983441751937951022640094154738645726447769371} a^{8} + \frac{239349421328244350171499325543357869565698735756478238564870}{1251232503323284489009398977403213247661577760396717583995093} a^{7} - \frac{27373078437002558026438610299256309674068555291358572065211789}{117615855312388741966883503875902045280188309477291452895538742} a^{6} + \frac{1715531174662054159989290779989237351123384061010036915882596}{5346175241472215543949250176177365694554014067149611495251761} a^{5} + \frac{48188159284779779401021625604067153845759977802316920899655447}{117615855312388741966883503875902045280188309477291452895538742} a^{4} + \frac{20578886270014891192489924691859859769434053197324960990672227}{58807927656194370983441751937951022640094154738645726447769371} a^{3} - \frac{23935571357753126158393825452182495512145695333874809467241900}{58807927656194370983441751937951022640094154738645726447769371} a^{2} + \frac{2647949437376549592093306545479729545890365538935712309618647}{5346175241472215543949250176177365694554014067149611495251761} a - \frac{15122724827991638797946893135057410178941153885179190541822045}{58807927656194370983441751937951022640094154738645726447769371}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  $12$
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:  \( 10194481907400 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\PSL(3,4)$ (as 21T67):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 20160
The 10 conjugacy class representatives for $\PSL(3,4)$
Character table for $\PSL(3,4)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Arithmetically equvalently siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.4.6.7$x^{4} + 2 x^{3} + 2 x^{2} + 2$$4$$1$$6$$A_4$$[2, 2]^{3}$
2.4.6.7$x^{4} + 2 x^{3} + 2 x^{2} + 2$$4$$1$$6$$A_4$$[2, 2]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
$11$11.7.6.1$x^{7} - 11$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
11.7.6.1$x^{7} - 11$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
11.7.6.1$x^{7} - 11$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
$67$$\Q_{67}$$x + 4$$1$$1$$0$Trivial$[\ ]$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.8.4.1$x^{8} + 17956 x^{4} - 300763 x^{2} + 80604484$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
67.8.4.1$x^{8} + 17956 x^{4} - 300763 x^{2} + 80604484$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$