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Properties

Label 21.21.8662050785...1889.1
Degree $21$
Signature $[21, 0]$
Discriminant $313^{14}$
Root discriminant $46.10$
Ramified prime $313$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_7:C_3$ (as 21T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-64, -1600, -13584, -42256, -10000, 166516, 200827, -158283, -317626, 58671, 230162, -9112, -94736, 2230, 23199, -1293, -3262, 329, 233, -33, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^20 - 33*x^19 + 233*x^18 + 329*x^17 - 3262*x^16 - 1293*x^15 + 23199*x^14 + 2230*x^13 - 94736*x^12 - 9112*x^11 + 230162*x^10 + 58671*x^9 - 317626*x^8 - 158283*x^7 + 200827*x^6 + 166516*x^5 - 10000*x^4 - 42256*x^3 - 13584*x^2 - 1600*x - 64)
 
gp: K = bnfinit(x^21 - 6*x^20 - 33*x^19 + 233*x^18 + 329*x^17 - 3262*x^16 - 1293*x^15 + 23199*x^14 + 2230*x^13 - 94736*x^12 - 9112*x^11 + 230162*x^10 + 58671*x^9 - 317626*x^8 - 158283*x^7 + 200827*x^6 + 166516*x^5 - 10000*x^4 - 42256*x^3 - 13584*x^2 - 1600*x - 64, 1)
 

Normalized defining polynomial

\( x^{21} - 6 x^{20} - 33 x^{19} + 233 x^{18} + 329 x^{17} - 3262 x^{16} - 1293 x^{15} + 23199 x^{14} + 2230 x^{13} - 94736 x^{12} - 9112 x^{11} + 230162 x^{10} + 58671 x^{9} - 317626 x^{8} - 158283 x^{7} + 200827 x^{6} + 166516 x^{5} - 10000 x^{4} - 42256 x^{3} - 13584 x^{2} - 1600 x - 64 \)

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:  \(86620507852136986313803229728551889=313^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.10$
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:  $313$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{13} + \frac{1}{12} a^{11} - \frac{1}{4} a^{10} + \frac{1}{12} a^{9} - \frac{1}{6} a^{8} - \frac{1}{12} a^{6} - \frac{1}{2} a^{5} - \frac{5}{12} a^{4} + \frac{5}{12} a^{3} + \frac{5}{12} a^{2} - \frac{1}{3}$, $\frac{1}{12} a^{14} + \frac{1}{12} a^{12} + \frac{1}{12} a^{10} - \frac{1}{6} a^{9} - \frac{1}{4} a^{8} + \frac{1}{6} a^{7} + \frac{1}{4} a^{6} - \frac{1}{6} a^{5} - \frac{1}{12} a^{4} + \frac{1}{6} a^{3} - \frac{1}{4} a^{2} + \frac{1}{6} a$, $\frac{1}{48} a^{15} - \frac{1}{48} a^{14} + \frac{1}{24} a^{12} + \frac{1}{8} a^{10} - \frac{5}{48} a^{9} + \frac{5}{24} a^{8} + \frac{1}{12} a^{7} + \frac{5}{48} a^{6} + \frac{19}{48} a^{5} - \frac{1}{48} a^{4} + \frac{23}{48} a^{3} - \frac{1}{4} a^{2} - \frac{5}{12} a - \frac{1}{6}$, $\frac{1}{48} a^{16} - \frac{1}{48} a^{14} - \frac{1}{24} a^{13} + \frac{1}{24} a^{12} + \frac{1}{24} a^{11} - \frac{11}{48} a^{10} + \frac{1}{48} a^{9} - \frac{1}{24} a^{8} + \frac{3}{16} a^{7} - \frac{5}{12} a^{6} - \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{5}{16} a^{3} - \frac{1}{12} a^{2} - \frac{1}{12} a + \frac{1}{6}$, $\frac{1}{144} a^{17} - \frac{1}{144} a^{15} - \frac{1}{72} a^{14} - \frac{1}{72} a^{13} + \frac{7}{72} a^{12} - \frac{5}{48} a^{11} - \frac{35}{144} a^{10} + \frac{1}{24} a^{9} + \frac{5}{144} a^{8} - \frac{1}{18} a^{7} - \frac{7}{72} a^{6} - \frac{5}{24} a^{5} - \frac{49}{144} a^{4} + \frac{1}{4} a^{3} - \frac{1}{3} a^{2} + \frac{7}{18} a + \frac{4}{9}$, $\frac{1}{144} a^{18} - \frac{1}{144} a^{16} + \frac{1}{144} a^{15} - \frac{5}{144} a^{14} + \frac{1}{72} a^{13} - \frac{1}{16} a^{12} - \frac{11}{144} a^{11} - \frac{1}{12} a^{10} - \frac{11}{72} a^{9} + \frac{5}{72} a^{8} + \frac{17}{72} a^{7} - \frac{13}{48} a^{6} - \frac{7}{36} a^{5} + \frac{7}{48} a^{4} - \frac{1}{48} a^{3} + \frac{17}{36} a^{2} - \frac{17}{36} a + \frac{1}{6}$, $\frac{1}{77472} a^{19} - \frac{17}{6456} a^{18} + \frac{257}{77472} a^{17} - \frac{101}{77472} a^{16} + \frac{673}{77472} a^{15} + \frac{497}{19368} a^{14} - \frac{1043}{25824} a^{13} - \frac{3719}{77472} a^{12} - \frac{135}{4304} a^{11} - \frac{5141}{38736} a^{10} + \frac{3631}{19368} a^{9} + \frac{3751}{19368} a^{8} + \frac{1567}{8608} a^{7} - \frac{1663}{19368} a^{6} + \frac{3045}{8608} a^{5} - \frac{593}{8608} a^{4} + \frac{3355}{38736} a^{3} - \frac{5951}{19368} a^{2} - \frac{386}{807} a + \frac{175}{1614}$, $\frac{1}{133627122440117568} a^{20} - \frac{74916987557}{33406780610029392} a^{19} - \frac{439974689972497}{133627122440117568} a^{18} - \frac{90685668287015}{44542374146705856} a^{17} + \frac{877764881029775}{133627122440117568} a^{16} + \frac{23477211222909}{3711864512225488} a^{15} - \frac{5168889787038853}{133627122440117568} a^{14} - \frac{3208829822013755}{133627122440117568} a^{13} + \frac{402369381927127}{8351695152507348} a^{12} + \frac{3425747083335863}{33406780610029392} a^{11} - \frac{430839593733611}{11135593536676464} a^{10} - \frac{12498832499450699}{66813561220058784} a^{9} + \frac{21563708192058079}{133627122440117568} a^{8} - \frac{7979209062190633}{33406780610029392} a^{7} + \frac{26709238078426973}{133627122440117568} a^{6} + \frac{2014628366775527}{44542374146705856} a^{5} + \frac{28597342543902733}{66813561220058784} a^{4} + \frac{3635469743627705}{11135593536676464} a^{3} + \frac{824554552990765}{8351695152507348} a^{2} + \frac{213463533899539}{927966128056372} a - \frac{44906663582095}{231991532014093}$

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:  \( 97964797589.8 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_7:C_3$ (as 21T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 21
The 5 conjugacy class representatives for $C_7:C_3$
Character table for $C_7:C_3$

Intermediate fields

3.3.97969.1, 7.7.9597924961.1 x7

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

Sibling fields

Degree 7 sibling: 7.7.9597924961.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/19.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
313Data not computed