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Properties

Label 21.1.86975280108...5488.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{33}\cdot 7^{40}\cdot 769^{7}$
Root discriminant $1108.49$
Ramified primes $2, 7, 769$
Class number Not computed
Class group Not computed
Galois group $S_3\times F_7$ (as 21T15)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-35473411152832, -50824408657344, -34923249923744, -11182890702928, 1078888675256, 1888661691252, 479101024822, -5603826919, -35791605276, -8204305753, 529827578, 517592649, 60618320, -10193785, -3258486, -84749, 68628, 7581, -602, -133, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 133*x^19 - 602*x^18 + 7581*x^17 + 68628*x^16 - 84749*x^15 - 3258486*x^14 - 10193785*x^13 + 60618320*x^12 + 517592649*x^11 + 529827578*x^10 - 8204305753*x^9 - 35791605276*x^8 - 5603826919*x^7 + 479101024822*x^6 + 1888661691252*x^5 + 1078888675256*x^4 - 11182890702928*x^3 - 34923249923744*x^2 - 50824408657344*x - 35473411152832)
 
gp: K = bnfinit(x^21 - 133*x^19 - 602*x^18 + 7581*x^17 + 68628*x^16 - 84749*x^15 - 3258486*x^14 - 10193785*x^13 + 60618320*x^12 + 517592649*x^11 + 529827578*x^10 - 8204305753*x^9 - 35791605276*x^8 - 5603826919*x^7 + 479101024822*x^6 + 1888661691252*x^5 + 1078888675256*x^4 - 11182890702928*x^3 - 34923249923744*x^2 - 50824408657344*x - 35473411152832, 1)
 

Normalized defining polynomial

\( x^{21} - 133 x^{19} - 602 x^{18} + 7581 x^{17} + 68628 x^{16} - 84749 x^{15} - 3258486 x^{14} - 10193785 x^{13} + 60618320 x^{12} + 517592649 x^{11} + 529827578 x^{10} - 8204305753 x^{9} - 35791605276 x^{8} - 5603826919 x^{7} + 479101024822 x^{6} + 1888661691252 x^{5} + 1078888675256 x^{4} - 11182890702928 x^{3} - 34923249923744 x^{2} - 50824408657344 x - 35473411152832 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(8697528010877856619486913745558327692042564236100084428584255488=2^{33}\cdot 7^{40}\cdot 769^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1108.49$
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:  $2, 7, 769$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{8} a^{9} + \frac{1}{8} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} + \frac{1}{16} a^{6} + \frac{3}{16} a^{5} - \frac{1}{8} a^{4}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{16} a^{7} + \frac{1}{16} a^{5}$, $\frac{1}{32} a^{12} - \frac{1}{32} a^{10} - \frac{1}{16} a^{9} - \frac{1}{32} a^{8} - \frac{1}{8} a^{7} + \frac{1}{32} a^{6} - \frac{1}{16} a^{5} + \frac{1}{4} a^{3}$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{11} + \frac{1}{32} a^{9} - \frac{3}{32} a^{7} + \frac{1}{16} a^{5}$, $\frac{1}{3136} a^{14} + \frac{5}{448} a^{13} - \frac{3}{448} a^{12} - \frac{1}{64} a^{11} - \frac{5}{448} a^{10} + \frac{27}{448} a^{9} + \frac{23}{448} a^{8} + \frac{173}{3136} a^{7} + \frac{13}{224} a^{6} - \frac{9}{112} a^{5} + \frac{3}{14} a^{4} - \frac{1}{2} a^{3} + \frac{3}{7} a^{2} - \frac{2}{7} a - \frac{1}{49}$, $\frac{1}{6272} a^{15} - \frac{5}{448} a^{13} - \frac{1}{64} a^{12} + \frac{1}{56} a^{11} - \frac{11}{448} a^{10} + \frac{15}{448} a^{9} - \frac{183}{3136} a^{8} + \frac{57}{896} a^{7} + \frac{31}{448} a^{6} + \frac{31}{224} a^{5} + \frac{3}{16} a^{4} - \frac{9}{56} a^{3} + \frac{3}{28} a^{2} - \frac{1}{98} a + \frac{5}{14}$, $\frac{1}{12544} a^{16} - \frac{1}{12544} a^{15} - \frac{1}{6272} a^{14} - \frac{3}{896} a^{12} - \frac{5}{896} a^{11} - \frac{1}{224} a^{10} + \frac{31}{3136} a^{9} + \frac{345}{12544} a^{8} + \frac{431}{12544} a^{7} - \frac{9}{896} a^{6} - \frac{27}{448} a^{5} - \frac{3}{32} a^{4} - \frac{41}{112} a^{3} - \frac{9}{392} a^{2} + \frac{16}{49} a + \frac{93}{196}$, $\frac{1}{25088} a^{17} + \frac{1}{25088} a^{15} - \frac{1}{12544} a^{14} + \frac{5}{1792} a^{13} - \frac{1}{224} a^{12} + \frac{51}{1792} a^{11} - \frac{39}{6272} a^{10} - \frac{93}{3584} a^{9} + \frac{61}{3136} a^{8} - \frac{59}{25088} a^{7} + \frac{145}{1792} a^{6} + \frac{55}{896} a^{5} - \frac{75}{448} a^{4} - \frac{361}{1568} a^{3} + \frac{29}{112} a^{2} - \frac{43}{392} a + \frac{37}{392}$, $\frac{1}{50176} a^{18} - \frac{1}{50176} a^{17} + \frac{1}{50176} a^{16} - \frac{3}{50176} a^{15} - \frac{1}{6272} a^{14} - \frac{45}{3584} a^{13} + \frac{11}{3584} a^{12} + \frac{349}{25088} a^{11} - \frac{47}{50176} a^{10} + \frac{915}{50176} a^{9} - \frac{1667}{50176} a^{8} + \frac{9}{50176} a^{7} + \frac{213}{3584} a^{6} + \frac{235}{1792} a^{5} + \frac{923}{6272} a^{4} + \frac{1159}{3136} a^{3} - \frac{121}{1568} a^{2} - \frac{23}{49} a + \frac{43}{784}$, $\frac{1}{351232} a^{19} + \frac{3}{351232} a^{18} + \frac{5}{351232} a^{17} - \frac{11}{351232} a^{16} + \frac{3}{43904} a^{15} + \frac{9}{175616} a^{14} - \frac{361}{25088} a^{13} - \frac{715}{175616} a^{12} + \frac{10417}{351232} a^{11} - \frac{9817}{351232} a^{10} + \frac{17585}{351232} a^{9} + \frac{15681}{351232} a^{8} - \frac{13661}{175616} a^{7} + \frac{491}{12544} a^{6} + \frac{7531}{43904} a^{5} - \frac{4289}{21952} a^{4} - \frac{1417}{10976} a^{3} - \frac{363}{1372} a^{2} + \frac{2139}{5488} a + \frac{117}{343}$, $\frac{1}{6250442006106025810285144287325726564186422035824355206594743200918018633935846970758947063808} a^{20} + \frac{5300134758442546209067573023457238762435009184140477585851120139334490874450618180397977}{6250442006106025810285144287325726564186422035824355206594743200918018633935846970758947063808} a^{19} - \frac{6424405075804917087459612029101767954564366441649289403959346869166892951776813145352391}{1562610501526506452571286071831431641046605508956088801648685800229504658483961742689736765952} a^{18} + \frac{1134820676387168769463396354752318580822345606263331424944757961177447747866173335611545}{63780020470469651125358615176793128205983898324738318434640236744061414631998438477132112896} a^{17} + \frac{186986771694101914968842371571444046245371513088472870790664117029893443301736596227421783}{6250442006106025810285144287325726564186422035824355206594743200918018633935846970758947063808} a^{16} - \frac{411418229092657025728972096640014663149310123362631423383819200178889538706068207858774757}{6250442006106025810285144287325726564186422035824355206594743200918018633935846970758947063808} a^{15} - \frac{209348494097479321818412182730242400036968965080885246694985998876108844270333799977251069}{3125221003053012905142572143662863282093211017912177603297371600459009316967923485379473531904} a^{14} - \frac{1986975466927401349600158559678614745222590704480911126911221176376826511128115627010362377}{390652625381626613142821517957857910261651377239022200412171450057376164620990435672434191488} a^{13} + \frac{1391900255748682502636246911294123269206852690332639857133181696263751959199484821424191591}{6250442006106025810285144287325726564186422035824355206594743200918018633935846970758947063808} a^{12} - \frac{78015383362689756888287423489433093137642011436717811998048888726507283270994214436551400849}{6250442006106025810285144287325726564186422035824355206594743200918018633935846970758947063808} a^{11} - \frac{2299367356089727841578234614132173058249827192270897534108526015148322565024084938762420839}{111615035823321889469377576559387974360471822068292057260620414302107475605997267334981197568} a^{10} + \frac{151210014452480485980731576291767600601272774078864825648470535599958574400311908369534151549}{3125221003053012905142572143662863282093211017912177603297371600459009316967923485379473531904} a^{9} - \frac{217723965563899329123022195089319380154567052076514325841318015130333664931807840250738505687}{6250442006106025810285144287325726564186422035824355206594743200918018633935846970758947063808} a^{8} - \frac{412356151254365291881227821373219525189657726575031320738602712418026222977308124692615441203}{6250442006106025810285144287325726564186422035824355206594743200918018633935846970758947063808} a^{7} + \frac{33380251747024860768302556775693840987577060801250554706169975393401902482258713734338915055}{3125221003053012905142572143662863282093211017912177603297371600459009316967923485379473531904} a^{6} + \frac{96096657515549548375321645019789651239800570908582402479608754246058058813237684814358078121}{1562610501526506452571286071831431641046605508956088801648685800229504658483961742689736765952} a^{5} + \frac{1301753657071537370551184959651484654869568857722700028775860330231775516578914815269962559}{781305250763253226285643035915715820523302754478044400824342900114752329241980871344868382976} a^{4} + \frac{2307483010754182204002989537853262796921387118872927284021005135673165774349693143702263193}{55807517911660944734688788279693987180235911034146028630310207151053737802998633667490598784} a^{3} + \frac{5888242687015295142027080589595881677826291616927622461444514619153134452223797655197076733}{15025100976216408197800827613763765779294283739962392323545055771437544793115016756632084288} a^{2} - \frac{879383600203362857032091237626361276942177773558895286549189281562919952975344375546384121}{3756275244054102049450206903440941444823570934990598080886263942859386198278754189158021072} a + \frac{21853068466248636317966018504452562445929968604477491486696896830604864078221693583968567703}{97663156345406653285705379489464477565412844309755550103042862514344041155247608918108547872}$

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

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $10$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times F_7$ (as 21T15):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 252
The 21 conjugacy class representatives for $S_3\times F_7$
Character table for $S_3\times F_7$ is not computed

Intermediate fields

3.1.43064.1, 7.1.6200896666048.4

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 42 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.14.27.213$x^{14} + 2 x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{8} + 4 x^{6} + 4 x^{5} + 4 x^{2} - 2$$14$$1$$27$$(C_7:C_3) \times C_2$$[3]_{7}^{3}$
$7$7.7.13.2$x^{7} + 105$$7$$1$$13$$F_7$$[13/6]_{6}$
7.14.27.55$x^{14} - 49 x^{13} - 147 x^{12} + 147 x^{11} - 49 x^{9} - 21 x^{7} - 98 x^{6} - 98 x^{5} + 147 x^{4} - 98 x^{3} - 49 x^{2} - 147 x + 140$$14$$1$$27$$F_7 \times C_2$$[13/6]_{6}^{2}$
769Data not computed