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Properties

Label 21.1.71333866771...5952.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{33}\cdot 3^{18}\cdot 7^{21}\cdot 11^{19}\cdot 13^{7}$
Root discriminant $1098.08$
Ramified primes $2, 3, 7, 11, 13$
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![83379248, -83922384, 42161840, -71119664, 88338768, -45490368, 22197616, -9539064, 3003280, -845936, 634284, 119084, -91518, 60690, -25267, 13041, -3927, 1337, -273, 63, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 63*x^19 - 273*x^18 + 1337*x^17 - 3927*x^16 + 13041*x^15 - 25267*x^14 + 60690*x^13 - 91518*x^12 + 119084*x^11 + 634284*x^10 - 845936*x^9 + 3003280*x^8 - 9539064*x^7 + 22197616*x^6 - 45490368*x^5 + 88338768*x^4 - 71119664*x^3 + 42161840*x^2 - 83922384*x + 83379248)
 
gp: K = bnfinit(x^21 - 7*x^20 + 63*x^19 - 273*x^18 + 1337*x^17 - 3927*x^16 + 13041*x^15 - 25267*x^14 + 60690*x^13 - 91518*x^12 + 119084*x^11 + 634284*x^10 - 845936*x^9 + 3003280*x^8 - 9539064*x^7 + 22197616*x^6 - 45490368*x^5 + 88338768*x^4 - 71119664*x^3 + 42161840*x^2 - 83922384*x + 83379248, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 63 x^{19} - 273 x^{18} + 1337 x^{17} - 3927 x^{16} + 13041 x^{15} - 25267 x^{14} + 60690 x^{13} - 91518 x^{12} + 119084 x^{11} + 634284 x^{10} - 845936 x^{9} + 3003280 x^{8} - 9539064 x^{7} + 22197616 x^{6} - 45490368 x^{5} + 88338768 x^{4} - 71119664 x^{3} + 42161840 x^{2} - 83922384 x + 83379248 \)

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:  \(7133386677177057321844488601370585790057054877370150192469245952=2^{33}\cdot 3^{18}\cdot 7^{21}\cdot 11^{19}\cdot 13^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1098.08$
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:  $2, 3, 7, 11, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{44} a^{14} + \frac{1}{44} a^{13} + \frac{9}{44} a^{12} + \frac{7}{44} a^{11} - \frac{5}{44} a^{10} + \frac{9}{44} a^{9} + \frac{1}{44} a^{8} - \frac{3}{44} a^{7} + \frac{3}{11} a^{6} - \frac{1}{22} a^{5} + \frac{3}{22} a^{4} - \frac{1}{11} a^{3} + \frac{5}{11} a^{2} + \frac{3}{11} a + \frac{3}{11}$, $\frac{1}{44} a^{15} + \frac{2}{11} a^{13} - \frac{1}{22} a^{12} + \frac{5}{22} a^{11} - \frac{2}{11} a^{10} - \frac{2}{11} a^{9} - \frac{1}{11} a^{8} - \frac{7}{44} a^{7} + \frac{2}{11} a^{6} + \frac{2}{11} a^{5} - \frac{5}{22} a^{4} - \frac{5}{11} a^{3} - \frac{2}{11} a^{2} - \frac{3}{11}$, $\frac{1}{44} a^{16} - \frac{5}{22} a^{13} + \frac{1}{11} a^{12} + \frac{1}{22} a^{11} + \frac{5}{22} a^{10} - \frac{5}{22} a^{9} + \frac{7}{44} a^{8} + \frac{5}{22} a^{7} - \frac{1}{2} a^{6} - \frac{4}{11} a^{5} + \frac{5}{11} a^{4} - \frac{5}{11} a^{3} + \frac{4}{11} a^{2} - \frac{5}{11} a - \frac{2}{11}$, $\frac{1}{88} a^{17} - \frac{1}{88} a^{16} - \frac{1}{88} a^{15} - \frac{1}{88} a^{14} - \frac{7}{88} a^{13} - \frac{7}{88} a^{12} + \frac{17}{88} a^{11} + \frac{9}{88} a^{10} + \frac{9}{44} a^{9} + \frac{2}{11} a^{8} - \frac{1}{11} a^{7} + \frac{5}{11} a^{6} + \frac{4}{11} a^{5} - \frac{5}{22} a^{4} + \frac{5}{22} a^{3} - \frac{3}{11} a^{2} + \frac{4}{11} a + \frac{5}{11}$, $\frac{1}{88} a^{18} - \frac{5}{44} a^{13} - \frac{1}{44} a^{12} + \frac{9}{44} a^{11} - \frac{9}{88} a^{10} - \frac{9}{44} a^{9} - \frac{1}{4} a^{8} + \frac{7}{44} a^{7} - \frac{9}{22} a^{6} - \frac{5}{22} a^{5} + \frac{3}{11} a^{4} - \frac{7}{22} a^{3} + \frac{1}{11} a^{2} + \frac{5}{11} a + \frac{1}{11}$, $\frac{1}{968} a^{19} + \frac{1}{242} a^{17} + \frac{1}{121} a^{16} + \frac{1}{484} a^{15} + \frac{1}{242} a^{14} - \frac{16}{121} a^{13} + \frac{7}{121} a^{12} + \frac{225}{968} a^{11} - \frac{2}{11} a^{9} - \frac{30}{121} a^{8} + \frac{4}{11} a^{7} + \frac{57}{242} a^{6} + \frac{103}{242} a^{5} + \frac{89}{242} a^{4} + \frac{1}{121} a^{3} - \frac{32}{121} a^{2} + \frac{58}{121} a + \frac{15}{121}$, $\frac{1}{19737654882299073654788400429216024718221280561105259666846946632} a^{20} - \frac{406212401167649102468468052864806422938760603610099773136945}{9868827441149536827394200214608012359110640280552629833423473316} a^{19} + \frac{53733458038608308543306559413163768338417957112720095838325381}{9868827441149536827394200214608012359110640280552629833423473316} a^{18} - \frac{31293658211005416672351322746521911248468732172084922133312985}{19737654882299073654788400429216024718221280561105259666846946632} a^{17} + \frac{210933284579120158488652228240483470970960821997776859086321069}{19737654882299073654788400429216024718221280561105259666846946632} a^{16} + \frac{207438908455395862977203490196518520560853264354781444206389741}{19737654882299073654788400429216024718221280561105259666846946632} a^{15} + \frac{1142724456477176161520865712856233320808976484324173542941221}{1794332262027188514071672766292365883474661869191387242440631512} a^{14} - \frac{297524548483209683287913362113095765088066773694861953669147539}{19737654882299073654788400429216024718221280561105259666846946632} a^{13} - \frac{2143122755354384830606139183673719811394394603451591549840579025}{9868827441149536827394200214608012359110640280552629833423473316} a^{12} + \frac{690668314759609696496944852802114724368378563535769031068053951}{19737654882299073654788400429216024718221280561105259666846946632} a^{11} + \frac{8676990462525263329316335713870284968017945010984379730028181}{1794332262027188514071672766292365883474661869191387242440631512} a^{10} - \frac{274323846630114845308315685513521313802272686831345629697678637}{4934413720574768413697100107304006179555320140276314916711736658} a^{9} + \frac{185894435139812530550256330178665141122575962860107780853928730}{2467206860287384206848550053652003089777660070138157458355868329} a^{8} - \frac{4014928838581870022766957843286086083573054663342382390440693093}{9868827441149536827394200214608012359110640280552629833423473316} a^{7} + \frac{554294993510513817751402334306783861384870782866145010431543426}{2467206860287384206848550053652003089777660070138157458355868329} a^{6} - \frac{560330252002534436924733015839027984702796064755296416993319238}{2467206860287384206848550053652003089777660070138157458355868329} a^{5} - \frac{346657844081721215389599913359509861254659956195166613305552773}{4934413720574768413697100107304006179555320140276314916711736658} a^{4} - \frac{19866997671457356268446714577640823535643083679301611738362673}{40780278682436102592538017415735588260787769754349710055468898} a^{3} - \frac{404995894855782451489474850153206989756694328506766770336147848}{2467206860287384206848550053652003089777660070138157458355868329} a^{2} + \frac{218295719261452345068713716306971993011315229208626603616847690}{2467206860287384206848550053652003089777660070138157458355868329} a + \frac{819440229274405923790339782230296696373713209314214082492868838}{2467206860287384206848550053652003089777660070138157458355868329}$

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

3.1.1144.1, 7.1.68069081958026688.23

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 R R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ $21$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$$[\ ]_{7}^{6}$
3.14.12.1$x^{14} - 3 x^{7} + 18$$7$$2$$12$$F_7$$[\ ]_{7}^{6}$
7Data not computed
$11$11.7.6.1$x^{7} - 11$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
11.14.13.2$x^{14} + 33$$14$$1$$13$$(C_7:C_3) \times C_2$$[\ ]_{14}^{3}$
$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$$[\ ]^{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$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}$