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Properties

Label 21.21.2341157840...4333.1
Degree $21$
Signature $[21, 0]$
Discriminant $229^{7}\cdot 577^{9}$
Root discriminant $93.32$
Ramified primes $229, 577$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3\times D_7$ (as 21T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-79, 1819, -14944, 47660, -5734, -253693, 293449, 335989, -530658, -191382, 397419, 45616, -155515, 600, 33963, -2503, -4101, 486, 251, -37, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^20 - 37*x^19 + 251*x^18 + 486*x^17 - 4101*x^16 - 2503*x^15 + 33963*x^14 + 600*x^13 - 155515*x^12 + 45616*x^11 + 397419*x^10 - 191382*x^9 - 530658*x^8 + 335989*x^7 + 293449*x^6 - 253693*x^5 - 5734*x^4 + 47660*x^3 - 14944*x^2 + 1819*x - 79)
 
gp: K = bnfinit(x^21 - 6*x^20 - 37*x^19 + 251*x^18 + 486*x^17 - 4101*x^16 - 2503*x^15 + 33963*x^14 + 600*x^13 - 155515*x^12 + 45616*x^11 + 397419*x^10 - 191382*x^9 - 530658*x^8 + 335989*x^7 + 293449*x^6 - 253693*x^5 - 5734*x^4 + 47660*x^3 - 14944*x^2 + 1819*x - 79, 1)
 

Normalized defining polynomial

\( x^{21} - 6 x^{20} - 37 x^{19} + 251 x^{18} + 486 x^{17} - 4101 x^{16} - 2503 x^{15} + 33963 x^{14} + 600 x^{13} - 155515 x^{12} + 45616 x^{11} + 397419 x^{10} - 191382 x^{9} - 530658 x^{8} + 335989 x^{7} + 293449 x^{6} - 253693 x^{5} - 5734 x^{4} + 47660 x^{3} - 14944 x^{2} + 1819 x - 79 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[21, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(234115784066285834308882767309928987624333=229^{7}\cdot 577^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $93.32$
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:  $229, 577$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{91} a^{19} + \frac{3}{7} a^{18} - \frac{33}{91} a^{17} + \frac{1}{91} a^{16} - \frac{17}{91} a^{15} + \frac{2}{7} a^{14} + \frac{6}{13} a^{13} - \frac{27}{91} a^{12} + \frac{8}{91} a^{11} - \frac{43}{91} a^{10} + \frac{1}{13} a^{9} + \frac{9}{91} a^{8} - \frac{31}{91} a^{7} + \frac{8}{91} a^{6} - \frac{33}{91} a^{5} + \frac{6}{13} a^{4} - \frac{8}{91} a^{3} - \frac{11}{91} a^{2} + \frac{3}{13} a - \frac{16}{91}$, $\frac{1}{905829385138595478196779689733733} a^{20} - \frac{1823821612589082681894073890229}{905829385138595478196779689733733} a^{19} - \frac{23208614430680792455558804923186}{905829385138595478196779689733733} a^{18} - \frac{353826130469171672067948610229517}{905829385138595478196779689733733} a^{17} - \frac{7856158675896547978587530982500}{905829385138595478196779689733733} a^{16} + \frac{83949170850192506202648723470427}{905829385138595478196779689733733} a^{15} - \frac{57131012533854256352235891431701}{129404197876942211170968527104819} a^{14} + \frac{220883297715244472413693667064560}{905829385138595478196779689733733} a^{13} - \frac{371206561109349206392755943171783}{905829385138595478196779689733733} a^{12} + \frac{61420634964023978886520469255412}{905829385138595478196779689733733} a^{11} - \frac{9301336814329372453144130410330}{129404197876942211170968527104819} a^{10} - \frac{28161489349768024101607140136491}{905829385138595478196779689733733} a^{9} - \frac{72178124273998061130113285930537}{905829385138595478196779689733733} a^{8} + \frac{411699968748636200494323103467589}{905829385138595478196779689733733} a^{7} + \frac{363252614451175326681680795634165}{905829385138595478196779689733733} a^{6} + \frac{2854094293888656879609996072050}{129404197876942211170968527104819} a^{5} + \frac{312965457421718685422769063274098}{905829385138595478196779689733733} a^{4} + \frac{17437103445977018335228735647858}{69679183472199652168983053056441} a^{3} + \frac{48559129147877909086201528027868}{129404197876942211170968527104819} a^{2} + \frac{305659786137768221551944205842109}{905829385138595478196779689733733} a + \frac{11892326137330872947773026665079}{129404197876942211170968527104819}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $20$
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:  \( 108385002485000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times D_7$ (as 21T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 84
The 15 conjugacy class representatives for $S_3\times D_7$
Character table for $S_3\times D_7$

Intermediate fields

3.3.229.1, 7.7.192100033.1

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 ${\href{/LocalNumberField/2.14.0.1}{14} }{,}\,{\href{/LocalNumberField/2.7.0.1}{7} }$ $21$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $21$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $21$ $21$ ${\href{/LocalNumberField/23.14.0.1}{14} }{,}\,{\href{/LocalNumberField/23.7.0.1}{7} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.14.0.1}{14} }{,}\,{\href{/LocalNumberField/31.7.0.1}{7} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/59.14.0.1}{14} }{,}\,{\href{/LocalNumberField/59.7.0.1}{7} }$

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
229Data not computed
577Data not computed