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Properties

Label 21.3.35485930488...7447.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,3^{18}\cdot 7^{17}\cdot 13^{14}$
Root discriminant $68.50$
Ramified primes $3, 7, 13$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3\times F_7$ (as 21T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1239967, -3755923, 4119359, -722371, -718864, -1333540, 1728873, -522437, 167386, -196980, 99374, -38866, 15522, -5396, 4105, -2535, 893, -175, 26, -10, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 10*x^19 + 26*x^18 - 175*x^17 + 893*x^16 - 2535*x^15 + 4105*x^14 - 5396*x^13 + 15522*x^12 - 38866*x^11 + 99374*x^10 - 196980*x^9 + 167386*x^8 - 522437*x^7 + 1728873*x^6 - 1333540*x^5 - 718864*x^4 - 722371*x^3 + 4119359*x^2 - 3755923*x + 1239967)
 
gp: K = bnfinit(x^21 - x^20 - 10*x^19 + 26*x^18 - 175*x^17 + 893*x^16 - 2535*x^15 + 4105*x^14 - 5396*x^13 + 15522*x^12 - 38866*x^11 + 99374*x^10 - 196980*x^9 + 167386*x^8 - 522437*x^7 + 1728873*x^6 - 1333540*x^5 - 718864*x^4 - 722371*x^3 + 4119359*x^2 - 3755923*x + 1239967, 1)
 

Normalized defining polynomial

\( x^{21} - x^{20} - 10 x^{19} + 26 x^{18} - 175 x^{17} + 893 x^{16} - 2535 x^{15} + 4105 x^{14} - 5396 x^{13} + 15522 x^{12} - 38866 x^{11} + 99374 x^{10} - 196980 x^{9} + 167386 x^{8} - 522437 x^{7} + 1728873 x^{6} - 1333540 x^{5} - 718864 x^{4} - 722371 x^{3} + 4119359 x^{2} - 3755923 x + 1239967 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-354859304882011896712926901003999617447=-\,3^{18}\cdot 7^{17}\cdot 13^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.50$
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:  $3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{12} a^{14} + \frac{1}{6} a^{12} - \frac{1}{6} a^{8} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{1}{12}$, $\frac{1}{12} a^{15} + \frac{1}{6} a^{13} - \frac{1}{6} a^{9} + \frac{1}{6} a^{7} - \frac{1}{2} a^{6} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{12} a - \frac{1}{2}$, $\frac{1}{12} a^{16} + \frac{1}{6} a^{12} - \frac{1}{6} a^{10} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{3}$, $\frac{1}{36} a^{17} - \frac{1}{36} a^{15} - \frac{1}{36} a^{14} - \frac{1}{6} a^{13} + \frac{1}{9} a^{12} - \frac{1}{18} a^{11} - \frac{1}{6} a^{10} + \frac{2}{9} a^{9} + \frac{2}{9} a^{8} + \frac{5}{18} a^{6} + \frac{1}{9} a^{5} - \frac{1}{2} a^{4} + \frac{5}{36} a^{3} + \frac{2}{9} a^{2} - \frac{5}{12} a + \frac{1}{36}$, $\frac{1}{144} a^{18} + \frac{1}{144} a^{17} - \frac{1}{144} a^{16} + \frac{1}{36} a^{15} - \frac{1}{144} a^{14} + \frac{5}{72} a^{13} - \frac{11}{72} a^{12} - \frac{13}{72} a^{11} - \frac{1}{9} a^{10} + \frac{11}{72} a^{9} + \frac{2}{9} a^{8} - \frac{2}{9} a^{7} + \frac{13}{72} a^{6} + \frac{11}{72} a^{5} - \frac{67}{144} a^{4} - \frac{17}{144} a^{3} - \frac{37}{144} a^{2} - \frac{7}{18} a - \frac{59}{144}$, $\frac{1}{432} a^{19} + \frac{1}{432} a^{18} + \frac{1}{144} a^{17} + \frac{1}{27} a^{16} + \frac{7}{432} a^{15} - \frac{1}{72} a^{14} + \frac{25}{216} a^{13} - \frac{41}{216} a^{12} + \frac{1}{9} a^{11} + \frac{23}{216} a^{10} - \frac{2}{27} a^{9} + \frac{2}{9} a^{8} + \frac{25}{216} a^{7} - \frac{41}{216} a^{6} + \frac{55}{144} a^{5} - \frac{113}{432} a^{4} - \frac{113}{432} a^{3} - \frac{1}{12} a^{2} - \frac{131}{432} a - \frac{13}{54}$, $\frac{1}{10794464132049250379284584798021552057817503848593097328} a^{20} + \frac{447237304335654533009271690456974236805670951374525}{490657460547693199058390218091888729900795629481504424} a^{19} - \frac{5461764949327095292870349333883583133711840446585813}{5397232066024625189642292399010776028908751924296548664} a^{18} - \frac{79029296334758640940125763014449889299009267755604159}{10794464132049250379284584798021552057817503848593097328} a^{17} + \frac{363104414888832669773290000959467041336631879170742081}{10794464132049250379284584798021552057817503848593097328} a^{16} + \frac{170407451417524352587194162773541571586889251525006269}{10794464132049250379284584798021552057817503848593097328} a^{15} - \frac{168606438740870334181761039893106758787665990446676189}{5397232066024625189642292399010776028908751924296548664} a^{14} + \frac{520952598466161986532860882592905979676937298500311705}{2698616033012312594821146199505388014454375962148274332} a^{13} - \frac{1145843884889459148721890632310490148502669399948854807}{5397232066024625189642292399010776028908751924296548664} a^{12} + \frac{78062631423746518740129615988609402280131105292946907}{490657460547693199058390218091888729900795629481504424} a^{11} + \frac{979279236465915237862299417949910492778490602117421123}{5397232066024625189642292399010776028908751924296548664} a^{10} + \frac{118362544038861399436575757797142465992739173489844543}{2698616033012312594821146199505388014454375962148274332} a^{9} - \frac{1043778495789240293351980894980548589235762770905048795}{5397232066024625189642292399010776028908751924296548664} a^{8} - \frac{265781658479528628261068464350237916557493995946499281}{1349308016506156297410573099752694007227187981074137166} a^{7} + \frac{1636009820232879976298359893647228291315293715536558679}{10794464132049250379284584798021552057817503848593097328} a^{6} + \frac{374735959165226323111026265648191940968497679933153133}{2698616033012312594821146199505388014454375962148274332} a^{5} + \frac{33149505796652177324381723146338944695998845547644237}{674654008253078148705286549876347003613593990537068583} a^{4} + \frac{1344389886231855070541553889145248522636759507476773797}{10794464132049250379284584798021552057817503848593097328} a^{3} - \frac{13550206896356998814197069413727870850120626985664989}{10794464132049250379284584798021552057817503848593097328} a^{2} - \frac{316894966627192787765417510059748242845405473313329671}{10794464132049250379284584798021552057817503848593097328} a - \frac{1062338223824157322897890822607277363060526346545244417}{5397232066024625189642292399010776028908751924296548664}$

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:  $11$
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:  \( 232645565732.53918 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times F_7$ (as 21T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 126
The 21 conjugacy class representatives for $C_3\times F_7$
Character table for $C_3\times F_7$ is not computed

Intermediate fields

3.3.8281.2, 7.1.7141592367.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 21 siblings: data not computed
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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$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$$[\ ]_{7}^{6}$
7Data not computed
$13$13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$