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Properties

Label 21.1.14064575590...5088.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{26}\cdot 3^{19}\cdot 7^{40}\cdot 601^{7}$
Root discriminant $2189.71$
Ramified primes $2, 3, 7, 601$
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![-30092867037324, -29486874188772, -14383246351572, -2830120486452, 754915945140, 753092808804, 145289526228, -12996844992, -13682621484, -2682084636, 145844244, 187055820, 24367560, -2372160, -1383859, -83237, 20055, 5327, -119, -63, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - 63*x^19 - 119*x^18 + 5327*x^17 + 20055*x^16 - 83237*x^15 - 1383859*x^14 - 2372160*x^13 + 24367560*x^12 + 187055820*x^11 + 145844244*x^10 - 2682084636*x^9 - 13682621484*x^8 - 12996844992*x^7 + 145289526228*x^6 + 753092808804*x^5 + 754915945140*x^4 - 2830120486452*x^3 - 14383246351572*x^2 - 29486874188772*x - 30092867037324)
 
gp: K = bnfinit(x^21 - 7*x^20 - 63*x^19 - 119*x^18 + 5327*x^17 + 20055*x^16 - 83237*x^15 - 1383859*x^14 - 2372160*x^13 + 24367560*x^12 + 187055820*x^11 + 145844244*x^10 - 2682084636*x^9 - 13682621484*x^8 - 12996844992*x^7 + 145289526228*x^6 + 753092808804*x^5 + 754915945140*x^4 - 2830120486452*x^3 - 14383246351572*x^2 - 29486874188772*x - 30092867037324, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} - 63 x^{19} - 119 x^{18} + 5327 x^{17} + 20055 x^{16} - 83237 x^{15} - 1383859 x^{14} - 2372160 x^{13} + 24367560 x^{12} + 187055820 x^{11} + 145844244 x^{10} - 2682084636 x^{9} - 13682621484 x^{8} - 12996844992 x^{7} + 145289526228 x^{6} + 753092808804 x^{5} + 754915945140 x^{4} - 2830120486452 x^{3} - 14383246351572 x^{2} - 29486874188772 x - 30092867037324 \)

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:  \(14064575590537072617183510833911332264598919604082378886625262816985088=2^{26}\cdot 3^{19}\cdot 7^{40}\cdot 601^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2189.71$
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:  $2, 3, 7, 601$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{4}$, $\frac{1}{1764} a^{14} + \frac{17}{252} a^{13} - \frac{5}{84} a^{12} + \frac{37}{252} a^{11} - \frac{1}{252} a^{10} + \frac{5}{84} a^{9} - \frac{41}{252} a^{8} + \frac{185}{1764} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{3}{14}$, $\frac{1}{1764} a^{15} - \frac{11}{126} a^{13} - \frac{13}{126} a^{12} - \frac{1}{7} a^{11} - \frac{17}{126} a^{10} + \frac{11}{126} a^{9} + \frac{13}{98} a^{8} - \frac{37}{252} a^{7} - \frac{1}{2} a^{6} + \frac{1}{3} a^{4} - \frac{2}{7} a - \frac{1}{2}$, $\frac{1}{1764} a^{16} - \frac{1}{21} a^{13} + \frac{1}{42} a^{12} + \frac{1}{7} a^{11} + \frac{1}{7} a^{10} - \frac{5}{147} a^{9} + \frac{11}{84} a^{8} + \frac{20}{63} a^{7} + \frac{1}{3} a^{5} - \frac{2}{7} a^{2} - \frac{1}{2} a$, $\frac{1}{5292} a^{17} - \frac{1}{5292} a^{16} + \frac{1}{5292} a^{14} + \frac{17}{108} a^{13} - \frac{23}{252} a^{12} + \frac{37}{756} a^{11} - \frac{319}{5292} a^{10} - \frac{16}{441} a^{9} + \frac{5}{42} a^{8} - \frac{275}{588} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} + \frac{17}{42} a^{3} + \frac{3}{7} a^{2} + \frac{1}{14}$, $\frac{1}{5292} a^{18} - \frac{1}{5292} a^{16} + \frac{1}{5292} a^{15} - \frac{4}{189} a^{13} - \frac{61}{378} a^{12} - \frac{143}{882} a^{11} + \frac{5}{756} a^{10} - \frac{58}{441} a^{9} - \frac{209}{1764} a^{8} - \frac{73}{252} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} - \frac{2}{21} a^{4} - \frac{1}{6} a^{3} + \frac{3}{7} a^{2} + \frac{1}{14} a - \frac{1}{2}$, $\frac{1}{74088} a^{19} + \frac{1}{74088} a^{18} - \frac{1}{74088} a^{17} - \frac{1}{24696} a^{16} + \frac{13}{74088} a^{15} + \frac{5}{74088} a^{14} + \frac{289}{3528} a^{13} - \frac{2195}{74088} a^{12} - \frac{95}{9261} a^{11} - \frac{1298}{9261} a^{10} + \frac{1577}{12348} a^{9} - \frac{101}{2058} a^{8} + \frac{5347}{12348} a^{7} + \frac{1}{84} a^{6} + \frac{43}{196} a^{5} - \frac{1}{147} a^{4} - \frac{131}{294} a^{3} - \frac{13}{49} a^{2} + \frac{3}{98} a + \frac{24}{49}$, $\frac{1}{57603475226149859235504215941252104238378685277851448701863818349505323117206558584711311849132056} a^{20} + \frac{266016765775657112936361740895116810303159597333624347795619443894506166171397611374873664085}{57603475226149859235504215941252104238378685277851448701863818349505323117206558584711311849132056} a^{19} + \frac{4152150905012211560820841025329017650698213126991959870912179499711704891961781522283493936477}{57603475226149859235504215941252104238378685277851448701863818349505323117206558584711311849132056} a^{18} - \frac{68850408224144196464083775390925195306870551670616634618122126810411186202244783539484283225}{8229067889449979890786316563036014891196955039693064100266259764215046159600936940673044549876008} a^{17} + \frac{5266446176946937235570432544420556743209512302954217978769522357369851262866288418013746159513}{19201158408716619745168071980417368079459561759283816233954606116501774372402186194903770616377352} a^{16} + \frac{9787163551492490668863250419241815111585539538519882876114331165253837005228035300043523968795}{57603475226149859235504215941252104238378685277851448701863818349505323117206558584711311849132056} a^{15} + \frac{15607000558075280020416035807646410887491752578239215645153272437139787429511057894091292153063}{57603475226149859235504215941252104238378685277851448701863818349505323117206558584711311849132056} a^{14} + \frac{2597855435427538492256435970091174243593416993632518694066787162359550118758194137712010014597009}{19201158408716619745168071980417368079459561759283816233954606116501774372402186194903770616377352} a^{13} - \frac{1219887688252013022878584565808401876397680461800207892082329940413951925310693598886231175956623}{28801737613074929617752107970626052119189342638925724350931909174752661558603279292355655924566028} a^{12} - \frac{4734712034495212945403182753817049442981091557869506694730615060936928950531817629742830119470929}{28801737613074929617752107970626052119189342638925724350931909174752661558603279292355655924566028} a^{11} + \frac{682881732430936586673842274591271840233147845874215642699235126393718518154176602532211899402971}{4114533944724989945393158281518007445598477519846532050133129882107523079800468470336522274938004} a^{10} - \frac{92777408212078192139470345895803510851563041595185164745199743250226028705014297253550108738117}{738506092642946913275695076169898772286906221510916008998254081403914398938545622880914254476052} a^{9} - \frac{451213375835977230608262484958898845595522424452296860159960149685980905283305443800246392681963}{9600579204358309872584035990208684039729780879641908116977303058250887186201093097451885308188676} a^{8} + \frac{213874199647956504424828052920015611212122711738674569238351323070509165040569432031159484464173}{1066731022706478874731559554467631559969975653293545346330811450916765242911232566383542812020964} a^{7} - \frac{219673426899378674128737262085883324026898453983559364775648448049919162035353912150246739976989}{457170438302776660599239809057556382844275279982948005570347764678613675533385385592946919437556} a^{6} + \frac{17911943289576734660792267705093804142013511310269967707003699776353686406406871693096279134769}{76195073050462776766539968176259397140712546663824667595057960779768945922230897598824486572926} a^{5} - \frac{18993019206595209588797695646506157009466093076847818679227927281479118544749624442025231970832}{38097536525231388383269984088129698570356273331912333797528980389884472961115448799412243286463} a^{4} + \frac{1887520575515163515761296518404260464790291582245833819708665100141627629421045419749475836923}{16327515653670595021401421752055585101581259999391000198940991595664774126192335199748104265627} a^{3} + \frac{11568577607951145148505088700014396869783769795357813972987431556107049382040365307494306506963}{38097536525231388383269984088129698570356273331912333797528980389884472961115448799412243286463} a^{2} - \frac{15278625477131870804369942585512898188644106314447893230235138645530815052052879332536449427622}{38097536525231388383269984088129698570356273331912333797528980389884472961115448799412243286463} a + \frac{58370511878478753725626300334593175120383334567882089814149002530926037303191003552812230986}{2930579732710106798713075699086899890027405640916333369040690799221882535470419138416326406651}$

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

3.1.50484.1, 7.1.4520453669548992.31

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $21$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.14.20.10$x^{14} + 2 x^{12} + 2 x^{11} + 4 x^{10} + 2 x^{8} - 2 x^{7} - 2 x^{6} + 4 x^{4} + 4 x^{3} - 2$$14$$1$$20$$(C_7:C_3) \times C_2$$[2]_{7}^{3}$
$3$3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
3.14.13.2$x^{14} + 3$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$
$7$7.7.13.1$x^{7} + 7$$7$$1$$13$$F_7$$[13/6]_{6}$
7.14.27.54$x^{14} - 98 x^{13} + 49 x^{12} - 49 x^{11} - 98 x^{10} + 49 x^{9} + 49 x^{8} - 133 x^{7} - 147 x^{6} + 98 x^{5} - 49 x^{4} + 147 x^{3} - 49 x^{2} + 98 x - 119$$14$$1$$27$$F_7$$[13/6]_{6}$
601Data not computed