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Properties

Label 21.21.1102590702...6464.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{14}\cdot 37^{7}\cdot 577^{9}$
Root discriminant $80.68$
Ramified primes $2, 37, 577$
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![-1, 50, 526, -3010, -17134, 52846, 79711, -188281, -134152, 233828, 99810, -141310, -38237, 46924, 7979, -8860, -894, 928, 49, -49, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 49*x^19 + 49*x^18 + 928*x^17 - 894*x^16 - 8860*x^15 + 7979*x^14 + 46924*x^13 - 38237*x^12 - 141310*x^11 + 99810*x^10 + 233828*x^9 - 134152*x^8 - 188281*x^7 + 79711*x^6 + 52846*x^5 - 17134*x^4 - 3010*x^3 + 526*x^2 + 50*x - 1)
 
gp: K = bnfinit(x^21 - x^20 - 49*x^19 + 49*x^18 + 928*x^17 - 894*x^16 - 8860*x^15 + 7979*x^14 + 46924*x^13 - 38237*x^12 - 141310*x^11 + 99810*x^10 + 233828*x^9 - 134152*x^8 - 188281*x^7 + 79711*x^6 + 52846*x^5 - 17134*x^4 - 3010*x^3 + 526*x^2 + 50*x - 1, 1)
 

Normalized defining polynomial

\( x^{21} - x^{20} - 49 x^{19} + 49 x^{18} + 928 x^{17} - 894 x^{16} - 8860 x^{15} + 7979 x^{14} + 46924 x^{13} - 38237 x^{12} - 141310 x^{11} + 99810 x^{10} + 233828 x^{9} - 134152 x^{8} - 188281 x^{7} + 79711 x^{6} + 52846 x^{5} - 17134 x^{4} - 3010 x^{3} + 526 x^{2} + 50 x - 1 \)

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:  \(11025907029010321740711464965433403326464=2^{14}\cdot 37^{7}\cdot 577^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $80.68$
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:  $2, 37, 577$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5} a^{15} + \frac{2}{5} a^{13} - \frac{1}{5} a^{12} + \frac{2}{5} a^{11} + \frac{2}{5} a^{10} - \frac{2}{5} a^{9} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{16} + \frac{2}{5} a^{14} - \frac{1}{5} a^{13} + \frac{2}{5} a^{12} + \frac{2}{5} a^{11} - \frac{2}{5} a^{10} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{2}{5} a^{6} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{5} a^{17} - \frac{1}{5} a^{14} - \frac{2}{5} a^{13} - \frac{1}{5} a^{12} - \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{18} - \frac{2}{5} a^{14} + \frac{1}{5} a^{13} - \frac{2}{5} a^{12} - \frac{2}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{4} - \frac{2}{5} a^{3} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{6235} a^{19} - \frac{102}{6235} a^{18} + \frac{304}{6235} a^{17} + \frac{52}{1247} a^{16} + \frac{334}{6235} a^{15} - \frac{6}{215} a^{14} - \frac{444}{1247} a^{13} + \frac{547}{6235} a^{12} - \frac{185}{1247} a^{11} + \frac{25}{1247} a^{10} - \frac{589}{6235} a^{9} + \frac{3058}{6235} a^{8} - \frac{2396}{6235} a^{7} - \frac{1628}{6235} a^{6} - \frac{8}{6235} a^{5} - \frac{847}{6235} a^{4} - \frac{2734}{6235} a^{3} - \frac{2046}{6235} a^{2} - \frac{2364}{6235} a + \frac{1986}{6235}$, $\frac{1}{204184296061287395567468684735} a^{20} - \frac{837079845023690882139432}{40836859212257479113493736947} a^{19} - \frac{4844603481458346167050304781}{204184296061287395567468684735} a^{18} - \frac{7084361845929954033058973247}{204184296061287395567468684735} a^{17} - \frac{2964412504216790984787150527}{204184296061287395567468684735} a^{16} + \frac{255504147357404410791044690}{3141296862481344547191825919} a^{15} + \frac{92493211754542862441766401482}{204184296061287395567468684735} a^{14} + \frac{9945461243567077478863454974}{204184296061287395567468684735} a^{13} + \frac{3011399429771015989546299211}{15706484312406722735959129595} a^{12} - \frac{54671980404428964402885703583}{204184296061287395567468684735} a^{11} - \frac{82095341317459148240599614108}{204184296061287395567468684735} a^{10} - \frac{30914133149078916819183996012}{204184296061287395567468684735} a^{9} - \frac{76884879517545534351106134886}{204184296061287395567468684735} a^{8} - \frac{62713876657309919279483387048}{204184296061287395567468684735} a^{7} + \frac{3968507483354638945007452003}{204184296061287395567468684735} a^{6} - \frac{5315764549569465371875333087}{40836859212257479113493736947} a^{5} + \frac{83374760785634696232127218332}{204184296061287395567468684735} a^{4} + \frac{98238612280002277911277050462}{204184296061287395567468684735} a^{3} - \frac{3260425006970806088853289304}{40836859212257479113493736947} a^{2} - \frac{69450494834301551539389595286}{204184296061287395567468684735} a - \frac{24218842052512329958091436448}{204184296061287395567468684735}$

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:  \( 30696176966900 \) (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

3.3.148.1, 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 R $21$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }{,}\,{\href{/LocalNumberField/17.7.0.1}{7} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }{,}\,{\href{/LocalNumberField/19.7.0.1}{7} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }{,}\,{\href{/LocalNumberField/23.7.0.1}{7} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.14.0.1}{14} }{,}\,{\href{/LocalNumberField/31.7.0.1}{7} }$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/59.14.0.1}{14} }{,}\,{\href{/LocalNumberField/59.7.0.1}{7} }$

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
37Data not computed
577Data not computed