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Properties

Label 21.1.18423050491...0000.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{18}\cdot 3^{18}\cdot 5^{19}\cdot 7^{21}\cdot 13^{7}\cdot 83^{7}$
Root discriminant $1430.52$
Ramified primes $2, 3, 5, 7, 13, 83$
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![-685324272, 1964250960, -2269756608, 1556164848, -1035586608, 722578416, -189202188, 138040805, -37440235, 2983120, -2665229, 966385, 87794, 8281, 19159, -1099, 434, 581, -161, 56, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 56*x^19 - 161*x^18 + 581*x^17 + 434*x^16 - 1099*x^15 + 19159*x^14 + 8281*x^13 + 87794*x^12 + 966385*x^11 - 2665229*x^10 + 2983120*x^9 - 37440235*x^8 + 138040805*x^7 - 189202188*x^6 + 722578416*x^5 - 1035586608*x^4 + 1556164848*x^3 - 2269756608*x^2 + 1964250960*x - 685324272)
 
gp: K = bnfinit(x^21 - 7*x^20 + 56*x^19 - 161*x^18 + 581*x^17 + 434*x^16 - 1099*x^15 + 19159*x^14 + 8281*x^13 + 87794*x^12 + 966385*x^11 - 2665229*x^10 + 2983120*x^9 - 37440235*x^8 + 138040805*x^7 - 189202188*x^6 + 722578416*x^5 - 1035586608*x^4 + 1556164848*x^3 - 2269756608*x^2 + 1964250960*x - 685324272, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 56 x^{19} - 161 x^{18} + 581 x^{17} + 434 x^{16} - 1099 x^{15} + 19159 x^{14} + 8281 x^{13} + 87794 x^{12} + 966385 x^{11} - 2665229 x^{10} + 2983120 x^{9} - 37440235 x^{8} + 138040805 x^{7} - 189202188 x^{6} + 722578416 x^{5} - 1035586608 x^{4} + 1556164848 x^{3} - 2269756608 x^{2} + 1964250960 x - 685324272 \)

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:  \(1842305049185776426372317709444210377390880236285000000000000000000=2^{18}\cdot 3^{18}\cdot 5^{19}\cdot 7^{21}\cdot 13^{7}\cdot 83^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1430.52$
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:  $2, 3, 5, 7, 13, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} + \frac{1}{4} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{20} a^{14} + \frac{1}{20} a^{13} - \frac{1}{10} a^{12} + \frac{1}{10} a^{11} - \frac{1}{20} a^{10} + \frac{3}{20} a^{9} - \frac{1}{10} a^{8} + \frac{1}{5} a^{7} + \frac{1}{20} a^{6} + \frac{7}{20} a^{5} - \frac{2}{5} a^{4} + \frac{1}{10} a^{3} - \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{20} a^{15} + \frac{1}{10} a^{13} - \frac{1}{20} a^{12} - \frac{3}{20} a^{11} + \frac{1}{5} a^{10} + \frac{1}{20} a^{8} - \frac{3}{20} a^{7} - \frac{1}{5} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{3}{10} a^{3} - \frac{1}{10} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{20} a^{16} + \frac{1}{10} a^{13} + \frac{1}{20} a^{12} + \frac{1}{10} a^{10} + \frac{1}{20} a^{8} - \frac{1}{10} a^{7} - \frac{1}{10} a^{6} - \frac{1}{5} a^{5} + \frac{1}{5} a^{3} + \frac{1}{10} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{20} a^{17} - \frac{1}{20} a^{13} - \frac{1}{20} a^{12} - \frac{1}{10} a^{11} + \frac{1}{10} a^{10} - \frac{1}{4} a^{9} - \frac{3}{20} a^{8} + \frac{1}{5} a^{6} - \frac{1}{5} a^{5} + \frac{1}{4} a^{4} + \frac{2}{5} a^{3} + \frac{1}{10} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{40} a^{18} - \frac{1}{40} a^{16} - \frac{1}{20} a^{13} - \frac{1}{8} a^{12} + \frac{1}{10} a^{11} - \frac{1}{5} a^{10} - \frac{1}{4} a^{9} + \frac{7}{40} a^{8} - \frac{1}{40} a^{6} - \frac{7}{20} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{200} a^{19} + \frac{1}{200} a^{18} + \frac{3}{200} a^{17} - \frac{3}{200} a^{16} + \frac{1}{50} a^{15} + \frac{1}{50} a^{14} - \frac{11}{200} a^{13} + \frac{17}{200} a^{12} + \frac{1}{25} a^{11} + \frac{9}{50} a^{10} - \frac{3}{40} a^{9} - \frac{7}{40} a^{8} - \frac{1}{8} a^{7} - \frac{1}{40} a^{6} + \frac{1}{4} a^{5} - \frac{17}{50} a^{4} - \frac{6}{25} a^{3} + \frac{9}{50} a^{2} - \frac{2}{25} a - \frac{4}{25}$, $\frac{1}{1407214456958250016903782920635183484746259577800879683319010308781555806600600} a^{20} + \frac{1008035084957156262081065431373526113486705628825367734228044550178701352011}{1407214456958250016903782920635183484746259577800879683319010308781555806600600} a^{19} - \frac{611750455792022748107348569643753625137934573536014019694523984722224880359}{175901807119781252112972865079397935593282447225109960414876288597694475825075} a^{18} + \frac{32919681473273060635070971527927466165204023300181617374277060046460137888017}{1407214456958250016903782920635183484746259577800879683319010308781555806600600} a^{17} + \frac{18051861832760306060259589008956074389512133006498303935741895983441923763139}{1407214456958250016903782920635183484746259577800879683319010308781555806600600} a^{16} + \frac{10745153571434604681223829856064023165762272999738593997220649332251594761897}{703607228479125008451891460317591742373129788900439841659505154390777903300300} a^{15} - \frac{25105376014650773190681849181797255124694203451922890844128553898426677500001}{1407214456958250016903782920635183484746259577800879683319010308781555806600600} a^{14} - \frac{33493110957039947365214012402169547646437043392841457026383140611989304388783}{1407214456958250016903782920635183484746259577800879683319010308781555806600600} a^{13} + \frac{147160195677856656625707229628650411497901685845708498357980900875159081968053}{1407214456958250016903782920635183484746259577800879683319010308781555806600600} a^{12} - \frac{74521242988761828249135632497930418329331224269212201682605060152179221966457}{703607228479125008451891460317591742373129788900439841659505154390777903300300} a^{11} + \frac{69651840082643382204534620559064139426240575761952642426232351808218367079919}{281442891391650003380756584127036696949251915560175936663802061756311161320120} a^{10} - \frac{4460521356395635997668156551118732250862570994595132852326162217135961687207}{56288578278330000676151316825407339389850383112035187332760412351262232264024} a^{9} - \frac{1192461572283822028030925832692525696056924755227376347688589730410235535741}{7036072284791250084518914603175917423731297889004398416595051543907779033003} a^{8} + \frac{13319211945735146666409752663417254031190781866944045569275434417370481911863}{281442891391650003380756584127036696949251915560175936663802061756311161320120} a^{7} + \frac{29670284320884570165390309274243494852554363206907417090751055235143200181351}{281442891391650003380756584127036696949251915560175936663802061756311161320120} a^{6} + \frac{79290884399481371401333661454464562912955201204447629278691025564560385283729}{175901807119781252112972865079397935593282447225109960414876288597694475825075} a^{5} + \frac{10577302275650723216771177863305006953842237984422534478510226347620789427381}{703607228479125008451891460317591742373129788900439841659505154390777903300300} a^{4} + \frac{161140702063296494962634727907131087121961685567374185533635519239668893724159}{351803614239562504225945730158795871186564894450219920829752577195388951650150} a^{3} + \frac{60443377524819211596771548666460286342242097117942066599126429060999703082913}{175901807119781252112972865079397935593282447225109960414876288597694475825075} a^{2} - \frac{5606983958003137478905843962733789443221794477860069135548480813999500512749}{175901807119781252112972865079397935593282447225109960414876288597694475825075} a - \frac{16090317699779068782968408036971616698361981689171263568468023049531692066998}{35180361423956250422594573015879587118656489445021992082975257719538895165015}$

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

3.1.5395.1, 7.1.600362847000000.31

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 R R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $21$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $21$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$[\ ]_{7}^{3}$
2.14.12.1$x^{14} - 2 x^{7} + 4$$7$$2$$12$$(C_7:C_3) \times C_2$$[\ ]_{7}^{6}$
$3$3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
3.14.12.1$x^{14} - 3 x^{7} + 18$$7$$2$$12$$F_7$$[\ ]_{7}^{6}$
$5$5.7.6.1$x^{7} - 5$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
5.14.13.1$x^{14} - 5$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$
7Data not computed
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_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}$
$83$$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.4.2.1$x^{4} + 249 x^{2} + 27556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
83.4.2.1$x^{4} + 249 x^{2} + 27556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
83.4.2.1$x^{4} + 249 x^{2} + 27556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$