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Properties

Label 21.1.23225248894...6048.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{33}\cdot 3^{19}\cdot 7^{17}$
Root discriminant $38.80$
Ramified primes $2, 3, 7$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
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![-1352, 7592, -19144, 27464, -23032, 7976, 6060, -10652, 7012, -1140, -1948, 1676, -774, 178, 283, -207, -31, 29, -1, 11, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 11*x^19 - x^18 + 29*x^17 - 31*x^16 - 207*x^15 + 283*x^14 + 178*x^13 - 774*x^12 + 1676*x^11 - 1948*x^10 - 1140*x^9 + 7012*x^8 - 10652*x^7 + 6060*x^6 + 7976*x^5 - 23032*x^4 + 27464*x^3 - 19144*x^2 + 7592*x - 1352)
 
gp: K = bnfinit(x^21 - 7*x^20 + 11*x^19 - x^18 + 29*x^17 - 31*x^16 - 207*x^15 + 283*x^14 + 178*x^13 - 774*x^12 + 1676*x^11 - 1948*x^10 - 1140*x^9 + 7012*x^8 - 10652*x^7 + 6060*x^6 + 7976*x^5 - 23032*x^4 + 27464*x^3 - 19144*x^2 + 7592*x - 1352, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 11 x^{19} - x^{18} + 29 x^{17} - 31 x^{16} - 207 x^{15} + 283 x^{14} + 178 x^{13} - 774 x^{12} + 1676 x^{11} - 1948 x^{10} - 1140 x^{9} + 7012 x^{8} - 10652 x^{7} + 6060 x^{6} + 7976 x^{5} - 23032 x^{4} + 27464 x^{3} - 19144 x^{2} + 7592 x - 1352 \)

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:  \(2322524889444393135652695649026048=2^{33}\cdot 3^{19}\cdot 7^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.80$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{16} - \frac{1}{4} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6}$, $\frac{1}{12} a^{18} - \frac{1}{12} a^{17} - \frac{1}{12} a^{16} + \frac{1}{12} a^{15} - \frac{1}{4} a^{14} - \frac{1}{4} a^{13} - \frac{1}{12} a^{12} + \frac{1}{12} a^{11} + \frac{1}{3} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{3} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{156} a^{19} + \frac{1}{156} a^{18} + \frac{1}{26} a^{17} + \frac{2}{39} a^{16} + \frac{1}{78} a^{15} + \frac{2}{13} a^{14} - \frac{7}{39} a^{13} + \frac{5}{39} a^{12} + \frac{1}{4} a^{11} + \frac{15}{52} a^{10} + \frac{5}{13} a^{9} + \frac{11}{26} a^{8} - \frac{7}{78} a^{7} - \frac{17}{39} a^{6} - \frac{7}{26} a^{5} + \frac{1}{39} a^{4} + \frac{1}{3} a^{3} - \frac{4}{13} a^{2} - \frac{16}{39} a - \frac{1}{3}$, $\frac{1}{1349933116221384844900415189670132} a^{20} + \frac{128899562074912135568049015890}{337483279055346211225103797417533} a^{19} + \frac{18514599655427677214950074584687}{449977705407128281633471729890044} a^{18} - \frac{459411226849999712961624324300}{112494426351782070408367932472511} a^{17} - \frac{42626116010484438431056550490706}{337483279055346211225103797417533} a^{16} - \frac{41898815309426918254922840629879}{674966558110692422450207594835066} a^{15} - \frac{32057973416076859604553453265399}{337483279055346211225103797417533} a^{14} + \frac{58972395546708073782439580893441}{337483279055346211225103797417533} a^{13} + \frac{70413467027224424433372678948677}{449977705407128281633471729890044} a^{12} - \frac{238939400124630745128460216890503}{674966558110692422450207594835066} a^{11} - \frac{144449053613510673806143824502441}{1349933116221384844900415189670132} a^{10} + \frac{101225738880848590563324150905195}{337483279055346211225103797417533} a^{9} + \frac{75477038983664411257234243586637}{224988852703564140816735864945022} a^{8} + \frac{54832686366589830416787876968111}{674966558110692422450207594835066} a^{7} + \frac{324148181246947986584350159102543}{674966558110692422450207594835066} a^{6} - \frac{315277597925814937325581022141899}{674966558110692422450207594835066} a^{5} - \frac{21954504914845042606936019344513}{337483279055346211225103797417533} a^{4} + \frac{13473712490419927581637764850390}{112494426351782070408367932472511} a^{3} + \frac{54366645675632163560370873607640}{112494426351782070408367932472511} a^{2} + \frac{66410842507754536128251705897387}{337483279055346211225103797417533} a - \frac{4757888123272304949734440779353}{25960252235026631632700292109041}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}$, which has order $3$ (assuming GRH)

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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 353864063.8162742 \) (assuming GRH)
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

3.1.1176.1, 7.1.784147392.1

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 ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $21$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$$[\ ]_{7}^{3}$
2.14.27.13$x^{14} - 4 x^{12} + 8 x^{11} - 6 x^{10} - 4 x^{9} - 6 x^{8} + 4 x^{7} + 6 x^{6} - 4 x^{5} + 4 x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 6$$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$$[\ ]_{7}^{6}$
3.14.13.1$x^{14} - 3$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$
7Data not computed