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Properties

Label 21.7.88764111739...6816.1
Degree $21$
Signature $[7, 7]$
Discriminant $-\,2^{20}\cdot 7^{12}\cdot 11^{19}$
Root discriminant $51.50$
Ramified primes $2, 7, 11$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $S_3\times C_7:C_3$ (as 21T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![266, -4026, 28698, -122288, 323368, -498694, 349094, 110816, -409604, 310564, -81018, -26452, 28358, -12932, 4800, -686, -382, 74, 58, -12, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 4*x^20 - 12*x^19 + 58*x^18 + 74*x^17 - 382*x^16 - 686*x^15 + 4800*x^14 - 12932*x^13 + 28358*x^12 - 26452*x^11 - 81018*x^10 + 310564*x^9 - 409604*x^8 + 110816*x^7 + 349094*x^6 - 498694*x^5 + 323368*x^4 - 122288*x^3 + 28698*x^2 - 4026*x + 266)
 
gp: K = bnfinit(x^21 - 4*x^20 - 12*x^19 + 58*x^18 + 74*x^17 - 382*x^16 - 686*x^15 + 4800*x^14 - 12932*x^13 + 28358*x^12 - 26452*x^11 - 81018*x^10 + 310564*x^9 - 409604*x^8 + 110816*x^7 + 349094*x^6 - 498694*x^5 + 323368*x^4 - 122288*x^3 + 28698*x^2 - 4026*x + 266, 1)
 

Normalized defining polynomial

\( x^{21} - 4 x^{20} - 12 x^{19} + 58 x^{18} + 74 x^{17} - 382 x^{16} - 686 x^{15} + 4800 x^{14} - 12932 x^{13} + 28358 x^{12} - 26452 x^{11} - 81018 x^{10} + 310564 x^{9} - 409604 x^{8} + 110816 x^{7} + 349094 x^{6} - 498694 x^{5} + 323368 x^{4} - 122288 x^{3} + 28698 x^{2} - 4026 x + 266 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-887641117397815509982736884947746816=-\,2^{20}\cdot 7^{12}\cdot 11^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.50$
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:  $2, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11} a^{12} + \frac{1}{11} a^{11} - \frac{1}{11} a^{10} - \frac{5}{11} a^{9} + \frac{3}{11} a^{8} + \frac{4}{11} a^{7} - \frac{5}{11} a^{6} - \frac{1}{11} a^{4} + \frac{1}{11} a^{3} - \frac{5}{11} a^{2} + \frac{4}{11} a + \frac{5}{11}$, $\frac{1}{33} a^{13} + \frac{1}{33} a^{12} - \frac{1}{33} a^{11} - \frac{16}{33} a^{10} + \frac{14}{33} a^{9} + \frac{5}{11} a^{8} - \frac{16}{33} a^{7} + \frac{1}{3} a^{6} + \frac{10}{33} a^{5} + \frac{1}{33} a^{4} - \frac{16}{33} a^{3} + \frac{5}{11} a^{2} + \frac{16}{33} a + \frac{1}{3}$, $\frac{1}{33} a^{14} + \frac{1}{33} a^{12} - \frac{4}{11} a^{11} - \frac{2}{11} a^{10} - \frac{14}{33} a^{9} + \frac{1}{3} a^{8} + \frac{2}{11} a^{7} - \frac{16}{33} a^{6} - \frac{3}{11} a^{5} + \frac{13}{33} a^{4} + \frac{1}{33} a^{3} - \frac{14}{33} a^{2} + \frac{7}{33} a + \frac{4}{33}$, $\frac{1}{33} a^{15} - \frac{1}{33} a^{12} + \frac{7}{33} a^{11} - \frac{10}{33} a^{10} + \frac{1}{11} a^{9} - \frac{2}{11} a^{8} + \frac{5}{11} a^{7} - \frac{14}{33} a^{6} + \frac{1}{11} a^{5} - \frac{4}{11} a^{4} + \frac{14}{33} a^{3} - \frac{2}{33} a^{2} + \frac{1}{11} a + \frac{16}{33}$, $\frac{1}{33} a^{16} - \frac{1}{33} a^{12} + \frac{13}{33} a^{11} - \frac{4}{33} a^{10} - \frac{13}{33} a^{9} + \frac{1}{11} a^{8} - \frac{7}{33} a^{6} - \frac{2}{33} a^{5} - \frac{3}{11} a^{4} + \frac{2}{11} a^{3} - \frac{1}{11} a^{2} - \frac{4}{33} a - \frac{1}{33}$, $\frac{1}{363} a^{17} - \frac{1}{121} a^{16} - \frac{1}{121} a^{15} - \frac{5}{363} a^{14} + \frac{2}{363} a^{13} - \frac{4}{363} a^{12} + \frac{5}{363} a^{11} + \frac{98}{363} a^{10} + \frac{19}{363} a^{9} - \frac{64}{363} a^{8} + \frac{83}{363} a^{7} - \frac{50}{121} a^{6} + \frac{10}{121} a^{5} - \frac{104}{363} a^{4} - \frac{38}{363} a^{3} + \frac{3}{11} a^{2} - \frac{23}{121} a - \frac{170}{363}$, $\frac{1}{1815} a^{18} - \frac{23}{1815} a^{16} - \frac{1}{605} a^{15} + \frac{4}{363} a^{14} + \frac{13}{1815} a^{13} - \frac{62}{1815} a^{12} - \frac{32}{605} a^{11} - \frac{391}{1815} a^{10} - \frac{7}{1815} a^{9} - \frac{68}{363} a^{8} + \frac{23}{165} a^{7} - \frac{46}{1815} a^{6} - \frac{146}{1815} a^{5} + \frac{52}{605} a^{4} - \frac{532}{1815} a^{3} - \frac{289}{1815} a^{2} + \frac{160}{363} a + \frac{524}{1815}$, $\frac{1}{16335} a^{19} - \frac{1}{5445} a^{18} - \frac{13}{16335} a^{17} + \frac{67}{5445} a^{16} - \frac{37}{5445} a^{15} + \frac{233}{16335} a^{14} - \frac{3}{605} a^{13} - \frac{133}{3267} a^{12} + \frac{5447}{16335} a^{11} + \frac{2347}{5445} a^{10} + \frac{2267}{5445} a^{9} + \frac{2026}{5445} a^{8} + \frac{1094}{3267} a^{7} - \frac{3268}{16335} a^{6} + \frac{2113}{5445} a^{5} - \frac{37}{1089} a^{4} - \frac{5728}{16335} a^{3} + \frac{503}{1815} a^{2} + \frac{5684}{16335} a + \frac{2063}{16335}$, $\frac{1}{222313148429753957227965061378245} a^{20} + \frac{4466823482448092491120433809}{222313148429753957227965061378245} a^{19} - \frac{580899445592939768908725656}{2887183745840960483480065732185} a^{18} - \frac{57787583178730844913595322048}{44462629685950791445593012275649} a^{17} + \frac{3539309437384620198705268129}{235252008920374557913190541141} a^{16} + \frac{40443319501112705990341261793}{4042057244177344676872092025059} a^{15} + \frac{490754102160047997091757719303}{44462629685950791445593012275649} a^{14} - \frac{900989872738173200394395865116}{222313148429753957227965061378245} a^{13} + \frac{1370004070568510802545833907786}{74104382809917985742655020459415} a^{12} + \frac{69965309933486695795494650392748}{222313148429753957227965061378245} a^{11} + \frac{776176214414626867138497966869}{2245587357876302598262273347255} a^{10} + \frac{3103344208909223722839359750573}{8233820312213109526961668939935} a^{9} - \frac{1793370746512341687826252191374}{222313148429753957227965061378245} a^{8} + \frac{176487208736017809716347248322}{2245587357876302598262273347255} a^{7} + \frac{84694827735866953623315564432026}{222313148429753957227965061378245} a^{6} + \frac{28146004441826961903125657096432}{74104382809917985742655020459415} a^{5} - \frac{2876642246661313193323263221261}{20210286220886723384360460125295} a^{4} + \frac{41733680658596535214826058064727}{222313148429753957227965061378245} a^{3} + \frac{18299388767215718096753404273642}{44462629685950791445593012275649} a^{2} + \frac{16818249844231863502864156581271}{222313148429753957227965061378245} a + \frac{864269657993175801378743405957}{31759021204250565318280723054035}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $13$
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:  \( 16693350374.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times C_7:C_3$ (as 21T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 126
The 15 conjugacy class representatives for $S_3\times C_7:C_3$
Character table for $S_3\times C_7:C_3$

Intermediate fields

3.1.44.1, 7.7.272225149504.1

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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{7}$ R R ${\href{/LocalNumberField/13.14.0.1}{14} }{,}\,{\href{/LocalNumberField/13.7.0.1}{7} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.14.0.1}{14} }{,}\,{\href{/LocalNumberField/29.7.0.1}{7} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.14.0.1}{14} }{,}\,{\href{/LocalNumberField/41.7.0.1}{7} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
2Data not computed
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$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}$