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Properties

Label 21.1.28507828261...1472.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{33}\cdot 3^{18}\cdot 7^{21}\cdot 1063^{7}$
Root discriminant $544.44$
Ramified primes $2, 3, 7, 1063$
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![1329014280576, -433958835072, -9805798464, 196647632832, 7269519264, -41439308064, 14603743728, -2718060368, -261657928, 387981832, -7171388, -25300156, 7422478, -51030, -322251, 76713, 973, -2359, 427, 7, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 7*x^19 + 427*x^18 - 2359*x^17 + 973*x^16 + 76713*x^15 - 322251*x^14 - 51030*x^13 + 7422478*x^12 - 25300156*x^11 - 7171388*x^10 + 387981832*x^9 - 261657928*x^8 - 2718060368*x^7 + 14603743728*x^6 - 41439308064*x^5 + 7269519264*x^4 + 196647632832*x^3 - 9805798464*x^2 - 433958835072*x + 1329014280576)
 
gp: K = bnfinit(x^21 - 7*x^20 + 7*x^19 + 427*x^18 - 2359*x^17 + 973*x^16 + 76713*x^15 - 322251*x^14 - 51030*x^13 + 7422478*x^12 - 25300156*x^11 - 7171388*x^10 + 387981832*x^9 - 261657928*x^8 - 2718060368*x^7 + 14603743728*x^6 - 41439308064*x^5 + 7269519264*x^4 + 196647632832*x^3 - 9805798464*x^2 - 433958835072*x + 1329014280576, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 7 x^{19} + 427 x^{18} - 2359 x^{17} + 973 x^{16} + 76713 x^{15} - 322251 x^{14} - 51030 x^{13} + 7422478 x^{12} - 25300156 x^{11} - 7171388 x^{10} + 387981832 x^{9} - 261657928 x^{8} - 2718060368 x^{7} + 14603743728 x^{6} - 41439308064 x^{5} + 7269519264 x^{4} + 196647632832 x^{3} - 9805798464 x^{2} - 433958835072 x + 1329014280576 \)

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:  \(2850782826128958048988669116723104997174540074367395561472=2^{33}\cdot 3^{18}\cdot 7^{21}\cdot 1063^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $544.44$
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:  $2, 3, 7, 1063$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{8} a^{9} + \frac{1}{8} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{10} + \frac{1}{16} a^{7} - \frac{1}{16} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{32} a^{12} - \frac{1}{32} a^{10} + \frac{1}{32} a^{8} + \frac{1}{32} a^{6} - \frac{1}{4} a^{5} - \frac{3}{16} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{11} + \frac{1}{32} a^{9} + \frac{1}{32} a^{7} + \frac{1}{16} a^{5} + \frac{1}{8} a^{3} - \frac{1}{4} a$, $\frac{1}{96} a^{14} + \frac{1}{96} a^{13} - \frac{1}{96} a^{11} - \frac{1}{24} a^{10} - \frac{1}{32} a^{9} - \frac{1}{48} a^{8} + \frac{1}{96} a^{7} - \frac{3}{32} a^{6} + \frac{1}{16} a^{5} + \frac{3}{16} a^{4} - \frac{1}{8} a^{3} + \frac{3}{8} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{576} a^{15} - \frac{1}{576} a^{14} + \frac{1}{576} a^{13} - \frac{7}{576} a^{12} + \frac{7}{576} a^{11} - \frac{25}{576} a^{10} + \frac{19}{576} a^{9} + \frac{23}{576} a^{8} + \frac{7}{144} a^{7} - \frac{7}{96} a^{6} + \frac{1}{24} a^{5} - \frac{11}{48} a^{4} - \frac{1}{3} a^{3} - \frac{7}{24} a^{2} + \frac{5}{12} a - \frac{1}{6}$, $\frac{1}{1152} a^{16} - \frac{1}{1152} a^{15} - \frac{5}{1152} a^{14} - \frac{13}{1152} a^{13} - \frac{11}{1152} a^{12} - \frac{19}{1152} a^{11} - \frac{11}{1152} a^{10} + \frac{41}{1152} a^{9} - \frac{25}{576} a^{8} + \frac{1}{12} a^{7} - \frac{1}{96} a^{6} + \frac{5}{48} a^{5} - \frac{11}{48} a^{4} - \frac{5}{24} a^{3} - \frac{5}{12} a^{2} - \frac{5}{24} a$, $\frac{1}{2304} a^{17} - \frac{1}{2304} a^{16} + \frac{1}{2304} a^{15} + \frac{5}{2304} a^{14} - \frac{17}{2304} a^{13} - \frac{25}{2304} a^{12} - \frac{29}{2304} a^{11} - \frac{25}{2304} a^{10} + \frac{25}{576} a^{9} + \frac{13}{384} a^{8} + \frac{1}{32} a^{7} - \frac{1}{24} a^{6} - \frac{1}{48} a^{5} + \frac{7}{48} a^{4} - \frac{7}{48} a^{3} + \frac{7}{48} a^{2} + \frac{1}{4} a + \frac{1}{8}$, $\frac{1}{4608} a^{18} - \frac{1}{4608} a^{17} + \frac{1}{4608} a^{16} + \frac{1}{4608} a^{15} + \frac{11}{4608} a^{14} - \frac{5}{4608} a^{13} - \frac{1}{4608} a^{12} + \frac{67}{4608} a^{11} + \frac{31}{576} a^{10} - \frac{35}{2304} a^{9} - \frac{17}{1152} a^{8} - \frac{5}{576} a^{7} - \frac{11}{96} a^{6} + \frac{1}{48} a^{5} + \frac{7}{96} a^{4} + \frac{17}{96} a^{3} + \frac{1}{3} a^{2} - \frac{13}{48} a + \frac{11}{24}$, $\frac{1}{9216} a^{19} - \frac{1}{9216} a^{18} - \frac{1}{9216} a^{17} - \frac{1}{9216} a^{16} + \frac{5}{9216} a^{15} - \frac{35}{9216} a^{14} - \frac{115}{9216} a^{13} - \frac{71}{9216} a^{12} + \frac{115}{4608} a^{11} + \frac{13}{288} a^{10} - \frac{1}{1152} a^{9} - \frac{31}{768} a^{8} - \frac{19}{288} a^{7} + \frac{11}{192} a^{6} - \frac{17}{192} a^{5} + \frac{5}{192} a^{4} + \frac{37}{96} a^{3} + \frac{5}{24} a^{2} + \frac{1}{4} a + \frac{19}{48}$, $\frac{1}{668602713033610191352523576701391780797768709748656046663207306748602259194932343668359382016} a^{20} - \frac{15673556158530227443698404800310224850280506678877654314101663389213894756853544366967133}{668602713033610191352523576701391780797768709748656046663207306748602259194932343668359382016} a^{19} - \frac{34495126665523268863632336602367691571945800560077690900653393402371333549761226429959813}{668602713033610191352523576701391780797768709748656046663207306748602259194932343668359382016} a^{18} + \frac{105924710867411818903674804322423172276905172873931488889402935421162497767047913400437171}{668602713033610191352523576701391780797768709748656046663207306748602259194932343668359382016} a^{17} + \frac{48157293117864077369784614408614345903729748049060563807643149034542514913114353821464385}{668602713033610191352523576701391780797768709748656046663207306748602259194932343668359382016} a^{16} + \frac{1970795408376594315626776621769379051751630613141152114976205645260481235596629755100161}{668602713033610191352523576701391780797768709748656046663207306748602259194932343668359382016} a^{15} - \frac{1138036112679872139894252656575827916586984684430447327185174115427123872849076884056760655}{668602713033610191352523576701391780797768709748656046663207306748602259194932343668359382016} a^{14} - \frac{109543184279275921712154247045708534023839685478859525192198869558869776755787536586275187}{668602713033610191352523576701391780797768709748656046663207306748602259194932343668359382016} a^{13} - \frac{798911646273702247326508345156517869948227390226984721384790494507866816915030144908753683}{334301356516805095676261788350695890398884354874328023331603653374301129597466171834179691008} a^{12} + \frac{1282720007975274620816154576065514482880236418393225977470044902775311291107891337572741885}{83575339129201273919065447087673972599721088718582005832900913343575282399366542958544922752} a^{11} + \frac{1194911611027168837855885932346220193337052023003475725354271778323878871270554676690347565}{83575339129201273919065447087673972599721088718582005832900913343575282399366542958544922752} a^{10} + \frac{2665568477093239263903096207175914001458608843892003422434934832261076945078825041439090853}{167150678258402547838130894175347945199442177437164011665801826687150564798733085917089845504} a^{9} + \frac{21789764217031132511338339573659054626479378711085952339040468137501305588224353155182067}{1741152898525026539980530147659874429160856014970458454852102361324485049986802978303019224} a^{8} - \frac{883179099665405907801750738741829623961924440484962329530277141891403095465728060926187375}{13929223188200212319844241181278995433286848119763667638816818890595880399894423826424153792} a^{7} + \frac{414085039795906984369792997102103032303086216444110670784485165677117195694343641987667977}{13929223188200212319844241181278995433286848119763667638816818890595880399894423826424153792} a^{6} + \frac{1159896613395630788002910935686080263908158604290811410794593257212271497046627515664243945}{13929223188200212319844241181278995433286848119763667638816818890595880399894423826424153792} a^{5} + \frac{1663295848712235441398731415582516621380372523779950609698056445413698899108089305267237601}{6964611594100106159922120590639497716643424059881833819408409445297940199947211913212076896} a^{4} - \frac{270079757506071869379303248362646243065218126961788492474961607287007376009574983323198995}{1741152898525026539980530147659874429160856014970458454852102361324485049986802978303019224} a^{3} + \frac{621822366877882866071069471994292787130745705828604518689660680643522940627305317918625193}{1741152898525026539980530147659874429160856014970458454852102361324485049986802978303019224} a^{2} - \frac{347864426575852717532402949022285231240275449087630325711380876083010548714600898070114753}{1160768599016684359987020098439916286107237343313638969901401574216323366657868652202012816} a + \frac{127190443252546954353772660048987941605317453648882264524922354731603742889550607885307383}{290192149754171089996755024609979071526809335828409742475350393554080841664467163050503204}$

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

3.1.8504.1, 7.1.38423222208.4

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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ $21$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$$[\ ]_{7}^{3}$
2.14.27.213$x^{14} + 2 x^{12} + 4 x^{11} - 2 x^{10} + 2 x^{8} + 4 x^{6} + 4 x^{5} + 4 x^{2} - 2$$14$$1$$27$$(C_7:C_3) \times C_2$$[3]_{7}^{3}$
$3$3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
7Data not computed
1063Data not computed