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Properties

Label 21.3.76553376717...1811.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,7^{14}\cdot 29^{3}\cdot 77351^{3}$
Root discriminant $29.56$
Ramified primes $7, 29, 77351$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T45

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-64, -288, -736, -1192, -1332, -876, -219, 243, 177, 60, -33, 137, 44, 37, -73, 33, -27, 34, -35, 19, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^20 + 19*x^19 - 35*x^18 + 34*x^17 - 27*x^16 + 33*x^15 - 73*x^14 + 37*x^13 + 44*x^12 + 137*x^11 - 33*x^10 + 60*x^9 + 177*x^8 + 243*x^7 - 219*x^6 - 876*x^5 - 1332*x^4 - 1192*x^3 - 736*x^2 - 288*x - 64)
 
gp: K = bnfinit(x^21 - 6*x^20 + 19*x^19 - 35*x^18 + 34*x^17 - 27*x^16 + 33*x^15 - 73*x^14 + 37*x^13 + 44*x^12 + 137*x^11 - 33*x^10 + 60*x^9 + 177*x^8 + 243*x^7 - 219*x^6 - 876*x^5 - 1332*x^4 - 1192*x^3 - 736*x^2 - 288*x - 64, 1)
 

Normalized defining polynomial

\( x^{21} - 6 x^{20} + 19 x^{19} - 35 x^{18} + 34 x^{17} - 27 x^{16} + 33 x^{15} - 73 x^{14} + 37 x^{13} + 44 x^{12} + 137 x^{11} - 33 x^{10} + 60 x^{9} + 177 x^{8} + 243 x^{7} - 219 x^{6} - 876 x^{5} - 1332 x^{4} - 1192 x^{3} - 736 x^{2} - 288 x - 64 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-7655337671796811085236857471811=-\,7^{14}\cdot 29^{3}\cdot 77351^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.56$
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:  $7, 29, 77351$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{12} + \frac{1}{8} a^{9} + \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} + \frac{1}{8} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{3}{8} a^{7} - \frac{1}{4} a^{6} + \frac{3}{8} a^{5} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{13} + \frac{1}{16} a^{11} - \frac{3}{16} a^{9} + \frac{3}{16} a^{8} - \frac{1}{16} a^{7} - \frac{7}{16} a^{6} - \frac{1}{4} a^{5} - \frac{1}{16} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{15} - \frac{1}{16} a^{13} - \frac{1}{16} a^{12} + \frac{1}{16} a^{11} + \frac{1}{16} a^{10} + \frac{1}{8} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{7}{16} a^{6} + \frac{7}{16} a^{5} - \frac{3}{16} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{32} a^{16} - \frac{1}{32} a^{14} + \frac{1}{32} a^{13} + \frac{1}{32} a^{12} - \frac{3}{32} a^{11} + \frac{1}{16} a^{9} - \frac{1}{4} a^{8} + \frac{5}{32} a^{7} + \frac{11}{32} a^{6} - \frac{9}{32} a^{5} - \frac{3}{8} a^{4} - \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{17} - \frac{1}{32} a^{15} - \frac{1}{32} a^{14} - \frac{1}{32} a^{13} + \frac{1}{32} a^{12} - \frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{5}{32} a^{8} + \frac{1}{32} a^{7} - \frac{7}{32} a^{6} + \frac{1}{4} a^{5} + \frac{3}{16} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{832} a^{18} + \frac{3}{832} a^{17} + \frac{5}{832} a^{16} - \frac{9}{416} a^{15} - \frac{11}{416} a^{14} + \frac{1}{32} a^{13} + \frac{5}{832} a^{12} + \frac{23}{208} a^{11} + \frac{31}{416} a^{10} + \frac{61}{832} a^{9} + \frac{75}{416} a^{8} - \frac{15}{32} a^{7} + \frac{303}{832} a^{6} - \frac{3}{416} a^{5} + \frac{21}{104} a^{4} - \frac{51}{104} a^{3} + \frac{1}{26} a^{2} - \frac{2}{13} a - \frac{3}{13}$, $\frac{1}{3328} a^{19} + \frac{11}{1664} a^{17} - \frac{7}{3328} a^{16} + \frac{29}{1664} a^{15} + \frac{23}{832} a^{14} - \frac{177}{3328} a^{13} + \frac{181}{3328} a^{12} - \frac{15}{208} a^{11} - \frac{333}{3328} a^{10} + \frac{643}{3328} a^{9} + \frac{165}{1664} a^{8} + \frac{1161}{3328} a^{7} - \frac{499}{3328} a^{6} - \frac{181}{832} a^{5} + \frac{19}{832} a^{4} + \frac{79}{416} a^{3} + \frac{45}{104} a^{2} - \frac{33}{104} a - \frac{17}{52}$, $\frac{1}{13312} a^{20} - \frac{1}{13312} a^{19} - \frac{1}{6656} a^{18} - \frac{205}{13312} a^{17} - \frac{159}{13312} a^{16} + \frac{77}{6656} a^{15} + \frac{259}{13312} a^{14} - \frac{29}{6656} a^{13} - \frac{749}{13312} a^{12} - \frac{121}{1024} a^{11} - \frac{71}{832} a^{10} + \frac{2175}{13312} a^{9} + \frac{67}{1024} a^{8} + \frac{469}{3328} a^{7} + \frac{3527}{13312} a^{6} - \frac{493}{1664} a^{5} + \frac{275}{3328} a^{4} - \frac{391}{1664} a^{3} - \frac{25}{208} a^{2} - \frac{9}{416} a - \frac{67}{208}$

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:  $11$
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:  \( 14306648.6019 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T45:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8232
The 55 conjugacy class representatives for t21n45 are not computed
Character table for t21n45 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\)

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

Sibling fields

Degree 28 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ $21$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.7.0.1}{7} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ $21$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.7.0.1$x^{7} - x + 3$$1$$7$$0$$C_7$$[\ ]^{7}$
29.7.0.1$x^{7} - x + 3$$1$$7$$0$$C_7$$[\ ]^{7}$
77351Data not computed