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Properties

Label 21.11.3442861570...4967.1
Degree $21$
Signature $[11, 5]$
Discriminant $-\,3^{9}\cdot 280909^{5}$
Root discriminant $31.75$
Ramified primes $3, 280909$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T38

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 45, -214, 128, 820, -928, -1669, 2506, 1934, -3941, -982, 3718, -214, -2100, 567, 665, -305, -93, 71, -1, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^20 - x^19 + 71*x^18 - 93*x^17 - 305*x^16 + 665*x^15 + 567*x^14 - 2100*x^13 - 214*x^12 + 3718*x^11 - 982*x^10 - 3941*x^9 + 1934*x^8 + 2506*x^7 - 1669*x^6 - 928*x^5 + 820*x^4 + 128*x^3 - 214*x^2 + 45*x - 1)
 
gp: K = bnfinit(x^21 - 6*x^20 - x^19 + 71*x^18 - 93*x^17 - 305*x^16 + 665*x^15 + 567*x^14 - 2100*x^13 - 214*x^12 + 3718*x^11 - 982*x^10 - 3941*x^9 + 1934*x^8 + 2506*x^7 - 1669*x^6 - 928*x^5 + 820*x^4 + 128*x^3 - 214*x^2 + 45*x - 1, 1)
 

Normalized defining polynomial

\( x^{21} - 6 x^{20} - x^{19} + 71 x^{18} - 93 x^{17} - 305 x^{16} + 665 x^{15} + 567 x^{14} - 2100 x^{13} - 214 x^{12} + 3718 x^{11} - 982 x^{10} - 3941 x^{9} + 1934 x^{8} + 2506 x^{7} - 1669 x^{6} - 928 x^{5} + 820 x^{4} + 128 x^{3} - 214 x^{2} + 45 x - 1 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[11, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-34428615704115413560094117894967=-\,3^{9}\cdot 280909^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.75$
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:  $3, 280909$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{18} + \frac{1}{6} a^{17} + \frac{1}{6} a^{16} - \frac{1}{6} a^{15} + \frac{1}{6} a^{14} - \frac{1}{6} a^{13} + \frac{1}{6} a^{12} - \frac{1}{3} a^{11} + \frac{1}{6} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{6} a^{7} - \frac{1}{2} a^{6} + \frac{1}{3} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{150} a^{19} - \frac{1}{50} a^{18} + \frac{6}{25} a^{17} + \frac{8}{75} a^{16} + \frac{11}{150} a^{15} - \frac{11}{150} a^{14} + \frac{19}{75} a^{13} - \frac{1}{2} a^{12} - \frac{9}{50} a^{11} + \frac{11}{30} a^{10} - \frac{3}{50} a^{9} - \frac{2}{75} a^{8} + \frac{4}{75} a^{7} + \frac{37}{75} a^{6} + \frac{71}{150} a^{5} + \frac{73}{150} a^{4} - \frac{43}{150} a^{3} - \frac{13}{75} a^{2} - \frac{14}{75} a + \frac{28}{75}$, $\frac{1}{150} a^{20} + \frac{1}{75} a^{18} + \frac{4}{25} a^{17} + \frac{17}{75} a^{16} - \frac{14}{75} a^{15} - \frac{2}{15} a^{14} - \frac{11}{150} a^{13} + \frac{23}{150} a^{12} - \frac{17}{50} a^{11} + \frac{28}{75} a^{10} + \frac{23}{50} a^{9} - \frac{9}{25} a^{8} + \frac{73}{150} a^{7} - \frac{7}{150} a^{6} - \frac{32}{75} a^{5} + \frac{1}{150} a^{4} - \frac{11}{30} a^{3} - \frac{28}{75} a^{2} - \frac{1}{50} a + \frac{34}{75}$

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:  $15$
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:  \( 130916585.101 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T38:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5040
The 15 conjugacy class representatives for t21n38
Character table for t21n38

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 7 sibling: 7.5.7584543.1
Degree 14 sibling: Deg 14
Degree 30 sibling: data not computed
Degree 35 sibling: 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 ${\href{/LocalNumberField/2.7.0.1}{7} }^{3}$ R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{3}$

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$3.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
280909Data not computed