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Properties

Label 21.1.29168672250...0000.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{33}\cdot 3^{19}\cdot 5^{19}\cdot 7^{21}\cdot 223^{7}$
Root discriminant $1462.16$
Ramified primes $2, 3, 5, 7, 223$
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![-97809091200, 7011849600, 69480331200, -23651409600, -45591991200, 33289250400, 3242337840, -2826925296, 938347368, 56321496, -28570668, 12744564, 1594194, -11802, -108919, -15323, 1449, 1421, 7, -21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - 21*x^19 + 7*x^18 + 1421*x^17 + 1449*x^16 - 15323*x^15 - 108919*x^14 - 11802*x^13 + 1594194*x^12 + 12744564*x^11 - 28570668*x^10 + 56321496*x^9 + 938347368*x^8 - 2826925296*x^7 + 3242337840*x^6 + 33289250400*x^5 - 45591991200*x^4 - 23651409600*x^3 + 69480331200*x^2 + 7011849600*x - 97809091200)
 
gp: K = bnfinit(x^21 - 7*x^20 - 21*x^19 + 7*x^18 + 1421*x^17 + 1449*x^16 - 15323*x^15 - 108919*x^14 - 11802*x^13 + 1594194*x^12 + 12744564*x^11 - 28570668*x^10 + 56321496*x^9 + 938347368*x^8 - 2826925296*x^7 + 3242337840*x^6 + 33289250400*x^5 - 45591991200*x^4 - 23651409600*x^3 + 69480331200*x^2 + 7011849600*x - 97809091200, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} - 21 x^{19} + 7 x^{18} + 1421 x^{17} + 1449 x^{16} - 15323 x^{15} - 108919 x^{14} - 11802 x^{13} + 1594194 x^{12} + 12744564 x^{11} - 28570668 x^{10} + 56321496 x^{9} + 938347368 x^{8} - 2826925296 x^{7} + 3242337840 x^{6} + 33289250400 x^{5} - 45591991200 x^{4} - 23651409600 x^{3} + 69480331200 x^{2} + 7011849600 x - 97809091200 \)

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:  \(2916867225003158090761647087922058466205473669120000000000000000000=2^{33}\cdot 3^{19}\cdot 5^{19}\cdot 7^{21}\cdot 223^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1462.16$
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:  $2, 3, 5, 7, 223$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{20} a^{8} - \frac{1}{10} a^{7} + \frac{1}{5} a^{6} + \frac{1}{10} a^{5} + \frac{1}{20} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{20} a^{9} - \frac{1}{4} a^{5} + \frac{1}{10} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{20} a^{10} - \frac{1}{4} a^{6} + \frac{1}{10} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{20} a^{11} - \frac{1}{4} a^{7} + \frac{1}{10} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{40} a^{12} - \frac{1}{40} a^{10} - \frac{1}{40} a^{8} + \frac{1}{10} a^{7} + \frac{1}{40} a^{6} - \frac{1}{10} a^{5} + \frac{1}{10} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{40} a^{13} - \frac{1}{40} a^{11} - \frac{1}{40} a^{9} + \frac{9}{40} a^{7} - \frac{1}{10} a^{5} - \frac{1}{10} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{600} a^{14} - \frac{1}{600} a^{13} - \frac{1}{100} a^{12} - \frac{1}{120} a^{11} - \frac{1}{60} a^{10} - \frac{1}{200} a^{9} - \frac{1}{300} a^{8} + \frac{113}{600} a^{7} - \frac{1}{8} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{1200} a^{15} - \frac{1}{1200} a^{14} + \frac{3}{400} a^{13} - \frac{1}{240} a^{12} - \frac{1}{48} a^{11} - \frac{1}{400} a^{10} - \frac{17}{1200} a^{9} - \frac{7}{1200} a^{8} - \frac{1}{4} a^{7} + \frac{1}{5} a^{6} + \frac{3}{20} a^{5} + \frac{1}{10} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2400} a^{16} - \frac{1}{2400} a^{15} - \frac{1}{2400} a^{14} - \frac{1}{96} a^{13} + \frac{1}{480} a^{12} + \frac{17}{2400} a^{11} + \frac{53}{2400} a^{10} + \frac{53}{2400} a^{9} - \frac{1}{240} a^{8} - \frac{13}{120} a^{7} + \frac{1}{10} a^{6} + \frac{1}{20} a^{5} + \frac{1}{10} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{12000} a^{17} - \frac{1}{400} a^{13} - \frac{3}{1000} a^{12} + \frac{1}{100} a^{11} - \frac{1}{40} a^{10} - \frac{1}{160} a^{9} + \frac{7}{600} a^{8} - \frac{37}{250} a^{7} - \frac{31}{200} a^{6} + \frac{1}{5} a^{5} - \frac{3}{20} a^{4} - \frac{1}{2} a + \frac{2}{5}$, $\frac{1}{24000} a^{18} - \frac{1}{24000} a^{17} - \frac{1}{4800} a^{16} - \frac{1}{4800} a^{15} + \frac{1}{4800} a^{14} - \frac{97}{8000} a^{13} + \frac{61}{24000} a^{12} + \frac{109}{4800} a^{11} - \frac{7}{800} a^{10} - \frac{1}{100} a^{9} - \frac{3}{2000} a^{8} - \frac{353}{3000} a^{7} + \frac{13}{200} a^{6} - \frac{1}{10} a^{5} + \frac{1}{5} a^{4} - \frac{1}{4} a^{3} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{72000} a^{19} - \frac{1}{72000} a^{18} - \frac{1}{24000} a^{17} - \frac{1}{14400} a^{16} + \frac{1}{14400} a^{15} - \frac{17}{24000} a^{14} - \frac{239}{72000} a^{13} - \frac{367}{72000} a^{12} + \frac{1}{2400} a^{11} + \frac{1}{75} a^{10} - \frac{151}{12000} a^{9} - \frac{17}{1000} a^{8} + \frac{197}{3000} a^{7} + \frac{1}{150} a^{6} + \frac{1}{30} a^{5} + \frac{1}{5} a^{4} - \frac{1}{2} a^{3} - \frac{1}{10} a^{2} + \frac{1}{10} a - \frac{1}{5}$, $\frac{1}{80553824288722713018914637548476994322080619411924962944684555040804980770672000} a^{20} - \frac{362174469969283478148171919435555206415935537994827791305318457775377101079}{80553824288722713018914637548476994322080619411924962944684555040804980770672000} a^{19} + \frac{901027380237963007558824133768060806693577281543574498568983076747605558177}{80553824288722713018914637548476994322080619411924962944684555040804980770672000} a^{18} + \frac{1716908540203482499333593904160500223781950596256791855485376411392610995109}{80553824288722713018914637548476994322080619411924962944684555040804980770672000} a^{17} + \frac{1238227902070713636507766708300518992813334626191744489329831667880615869751}{16110764857744542603782927509695398864416123882384992588936911008160996154134400} a^{16} - \frac{6269187888267671881515652503880787928963413015362771634085539311281457795121}{80553824288722713018914637548476994322080619411924962944684555040804980770672000} a^{15} - \frac{24534831006628631505416185298427147226902891741569738068780888083729275671601}{80553824288722713018914637548476994322080619411924962944684555040804980770672000} a^{14} + \frac{238868297593194788537439678366905364653897304443645866294024844720524113629563}{80553824288722713018914637548476994322080619411924962944684555040804980770672000} a^{13} - \frac{84095024315853652570417223760403431522582021510937549717143791919233766365293}{10069228036090339127364329693559624290260077426490620368085569380100622596334000} a^{12} + \frac{18070930647893549892379409351119683479913870262676393364475955625438276242449}{1342563738145378550315243959141283238701343656865416049078075917346749679511200} a^{11} + \frac{32217293121456270469702714171426018136438485169423129638442525837152434752117}{6712818690726892751576219795706416193506718284327080245390379586733748397556000} a^{10} + \frac{59064962492429179010415126392410002435674017614254788703662696033365450305197}{6712818690726892751576219795706416193506718284327080245390379586733748397556000} a^{9} + \frac{72704954933954649097619918407473704904607352163804807934275555192371881103407}{3356409345363446375788109897853208096753359142163540122695189793366874198778000} a^{8} + \frac{55426305696160428731395382531538812824450139568464567029593139058784499300679}{559401557560574395964684982975534682792226523693923353782531632227812366463000} a^{7} + \frac{8924610049135442156878366947639098679005448757845256099614316829451576417598}{41955116817043079697351373723165101209416989277044251533689872417085927484725} a^{6} + \frac{1299303052485739700534977837172600808464119324977196481084741102376729213659}{13425637381453785503152439591412832387013436568654160490780759173467496795112} a^{5} + \frac{473235943117557287986408032050933579036877736979375272404473544590758314067}{5594015575605743959646849829755346827922265236939233537825316322278123664630} a^{4} - \frac{1809347845873647788674047075125459701695009919143158733710631184387419391673}{5594015575605743959646849829755346827922265236939233537825316322278123664630} a^{3} - \frac{328696072290838534797138909206381670444769502137362377613161721044400989133}{5594015575605743959646849829755346827922265236939233537825316322278123664630} a^{2} - \frac{1319142984049044449875673874465396209202838582505952581278852608471032578531}{5594015575605743959646849829755346827922265236939233537825316322278123664630} a + \frac{1013938941656377870911705267046390744985992483925089870350226883989840970084}{2797007787802871979823424914877673413961132618469616768912658161139061832315}$

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

3.1.26760.3, 7.1.600362847000000.17

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 ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ $21$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.14.27.121$x^{14} - 4 x^{13} - 4 x^{12} + 8 x^{11} + 8 x^{10} - 6 x^{8} + 8 x^{7} + 6 x^{6} - 4 x^{5} - 2 x^{4} - 4 x^{3} + 4 x^{2} + 8 x + 6$$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$$[\ ]_{7}^{6}$
3.14.13.2$x^{14} + 3$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$
$5$5.7.6.1$x^{7} - 5$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
5.14.13.2$x^{14} + 10$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$
7Data not computed
223Data not computed