↑

Properties

Label 21.21.4793264958...1728.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 7^{35}\cdot 13^{6}$
Root discriminant $96.56$
Ramified primes $2, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T31

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17576, -108836, -679042, -498043, 1656200, 2084446, -1388660, -2522507, 542556, 1541932, -106400, -550564, 10052, 120442, -362, -16198, 0, 1295, 0, -56, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 56*x^19 + 1295*x^17 - 16198*x^15 - 362*x^14 + 120442*x^13 + 10052*x^12 - 550564*x^11 - 106400*x^10 + 1541932*x^9 + 542556*x^8 - 2522507*x^7 - 1388660*x^6 + 2084446*x^5 + 1656200*x^4 - 498043*x^3 - 679042*x^2 - 108836*x + 17576)
 
gp: K = bnfinit(x^21 - 56*x^19 + 1295*x^17 - 16198*x^15 - 362*x^14 + 120442*x^13 + 10052*x^12 - 550564*x^11 - 106400*x^10 + 1541932*x^9 + 542556*x^8 - 2522507*x^7 - 1388660*x^6 + 2084446*x^5 + 1656200*x^4 - 498043*x^3 - 679042*x^2 - 108836*x + 17576, 1)
 

Normalized defining polynomial

\( x^{21} - 56 x^{19} + 1295 x^{17} - 16198 x^{15} - 362 x^{14} + 120442 x^{13} + 10052 x^{12} - 550564 x^{11} - 106400 x^{10} + 1541932 x^{9} + 542556 x^{8} - 2522507 x^{7} - 1388660 x^{6} + 2084446 x^{5} + 1656200 x^{4} - 498043 x^{3} - 679042 x^{2} - 108836 x + 17576 \)

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:  \(479326495885529228039730385744116026441728=2^{18}\cdot 7^{35}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $96.56$
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:  $2, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{2}$, $\frac{1}{52} a^{17} + \frac{9}{52} a^{15} + \frac{2}{13} a^{13} - \frac{1}{2} a^{11} + \frac{1}{26} a^{10} - \frac{4}{13} a^{9} - \frac{5}{26} a^{8} + \frac{3}{13} a^{7} + \frac{9}{26} a^{6} - \frac{6}{13} a^{5} + \frac{7}{26} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{52} a^{18} + \frac{9}{52} a^{16} + \frac{2}{13} a^{14} - \frac{1}{2} a^{12} + \frac{1}{26} a^{11} - \frac{4}{13} a^{10} - \frac{5}{26} a^{9} + \frac{3}{13} a^{8} + \frac{9}{26} a^{7} - \frac{6}{13} a^{6} + \frac{7}{26} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{104} a^{19} - \frac{21}{104} a^{15} - \frac{1}{2} a^{14} + \frac{3}{52} a^{13} - \frac{25}{52} a^{12} + \frac{5}{52} a^{11} - \frac{7}{26} a^{10} - \frac{1}{2} a^{9} - \frac{6}{13} a^{8} - \frac{7}{26} a^{7} - \frac{11}{26} a^{6} + \frac{21}{104} a^{5} + \frac{1}{26} a^{4} - \frac{1}{4} a^{3} - \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{6073224152832683168133567474368} a^{20} + \frac{928501163928567535240133063}{233585544339718583389752595168} a^{19} + \frac{4358840162655102597717633427}{1518306038208170792033391868592} a^{18} + \frac{242648273732335009061581079}{58396386084929645847438148792} a^{17} + \frac{244674127482310475876964582111}{6073224152832683168133567474368} a^{16} + \frac{19228788255950221337722626577}{233585544339718583389752595168} a^{15} + \frac{98757421042049699847788454807}{233585544339718583389752595168} a^{14} + \frac{274553251816145913789322064285}{3036612076416341584066783737184} a^{13} - \frac{1028403999212195139058276310437}{3036612076416341584066783737184} a^{12} + \frac{260068200630078513993346393681}{759153019104085396016695934296} a^{11} - \frac{164642894071478158513946816629}{1518306038208170792033391868592} a^{10} + \frac{146626780501043122402716433605}{759153019104085396016695934296} a^{9} - \frac{276435165043493571621124642497}{1518306038208170792033391868592} a^{8} + \frac{385317459611977072401565468641}{1518306038208170792033391868592} a^{7} + \frac{203691864970963524458511063969}{467171088679437166779505190336} a^{6} + \frac{91361593966432144268623502041}{233585544339718583389752595168} a^{5} - \frac{92361324147446883358575542191}{233585544339718583389752595168} a^{4} + \frac{2425255695341801755043075773}{8984059397681483976528945968} a^{3} + \frac{6765319950940982241712454357}{35936237590725935906115783872} a^{2} - \frac{4271521285153736111270196957}{8984059397681483976528945968} a - \frac{1656778332791290168287408031}{8984059397681483976528945968}$

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:  \( 268570252786000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T31:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2058
The 71 conjugacy class representatives for t21n31 are not computed
Character table for t21n31 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 21 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 R $21$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ $21$ $21$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ $21$ $21$ $21$

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
$2$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
7Data not computed
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.7.6.1$x^{7} - 13$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$