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Properties

Label 21.21.7104507042...6656.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 7^{14}\cdot 43^{12}$
Root discriminant $56.86$
Ramified primes $2, 7, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times C_7:C_3$ (as 21T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![104, 12440, -105968, 128848, 470904, -803144, -663064, 1451596, 302096, -1188672, 77632, 494124, -111352, -105928, 37382, 10446, -5560, -210, 362, -26, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 8*x^20 - 26*x^19 + 362*x^18 - 210*x^17 - 5560*x^16 + 10446*x^15 + 37382*x^14 - 105928*x^13 - 111352*x^12 + 494124*x^11 + 77632*x^10 - 1188672*x^9 + 302096*x^8 + 1451596*x^7 - 663064*x^6 - 803144*x^5 + 470904*x^4 + 128848*x^3 - 105968*x^2 + 12440*x + 104)
 
gp: K = bnfinit(x^21 - 8*x^20 - 26*x^19 + 362*x^18 - 210*x^17 - 5560*x^16 + 10446*x^15 + 37382*x^14 - 105928*x^13 - 111352*x^12 + 494124*x^11 + 77632*x^10 - 1188672*x^9 + 302096*x^8 + 1451596*x^7 - 663064*x^6 - 803144*x^5 + 470904*x^4 + 128848*x^3 - 105968*x^2 + 12440*x + 104, 1)
 

Normalized defining polynomial

\( x^{21} - 8 x^{20} - 26 x^{19} + 362 x^{18} - 210 x^{17} - 5560 x^{16} + 10446 x^{15} + 37382 x^{14} - 105928 x^{13} - 111352 x^{12} + 494124 x^{11} + 77632 x^{10} - 1188672 x^{9} + 302096 x^{8} + 1451596 x^{7} - 663064 x^{6} - 803144 x^{5} + 470904 x^{4} + 128848 x^{3} - 105968 x^{2} + 12440 x + 104 \)

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:  \(7104507042655360857983379370527686656=2^{18}\cdot 7^{14}\cdot 43^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.86$
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:  $2, 7, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{14}$, $\frac{1}{28} a^{15} + \frac{1}{14} a^{14} - \frac{1}{28} a^{13} + \frac{1}{7} a^{12} - \frac{3}{14} a^{11} - \frac{1}{14} a^{10} - \frac{1}{7} a^{9} - \frac{1}{14} a^{8} - \frac{1}{7} a^{7} + \frac{1}{14} a^{6} + \frac{2}{7} a^{5} - \frac{3}{7} a^{4} - \frac{1}{7} a^{3} - \frac{1}{7} a^{2} - \frac{1}{7} a - \frac{2}{7}$, $\frac{1}{28} a^{16} + \frac{1}{14} a^{14} - \frac{1}{28} a^{13} - \frac{1}{7} a^{11} + \frac{3}{14} a^{9} - \frac{1}{7} a^{7} - \frac{5}{14} a^{6} - \frac{2}{7} a^{4} + \frac{1}{7} a^{3} + \frac{1}{7} a^{2} - \frac{3}{7}$, $\frac{1}{28} a^{17} + \frac{1}{14} a^{14} + \frac{1}{14} a^{13} + \frac{1}{14} a^{12} - \frac{1}{14} a^{11} - \frac{1}{7} a^{10} - \frac{3}{14} a^{9} - \frac{1}{14} a^{7} - \frac{1}{7} a^{6} + \frac{1}{7} a^{5} + \frac{3}{7} a^{3} + \frac{2}{7} a^{2} - \frac{1}{7} a - \frac{3}{7}$, $\frac{1}{980} a^{18} + \frac{1}{245} a^{17} + \frac{4}{245} a^{16} - \frac{1}{980} a^{15} + \frac{2}{245} a^{14} + \frac{11}{980} a^{13} + \frac{9}{245} a^{12} + \frac{9}{49} a^{11} + \frac{9}{49} a^{10} - \frac{1}{35} a^{9} + \frac{57}{245} a^{8} + \frac{73}{490} a^{7} + \frac{13}{98} a^{6} + \frac{103}{245} a^{5} - \frac{34}{245} a^{4} + \frac{68}{245} a^{3} + \frac{96}{245} a^{2} + \frac{59}{245} a - \frac{89}{245}$, $\frac{1}{1960} a^{19} - \frac{3}{196} a^{16} + \frac{3}{490} a^{15} - \frac{1}{10} a^{14} + \frac{101}{980} a^{13} + \frac{9}{490} a^{12} + \frac{15}{98} a^{11} + \frac{29}{245} a^{10} + \frac{3}{98} a^{9} - \frac{69}{490} a^{8} - \frac{13}{245} a^{7} - \frac{237}{490} a^{6} - \frac{201}{490} a^{5} + \frac{67}{245} a^{4} + \frac{52}{245} a^{3} + \frac{20}{49} a^{2} - \frac{8}{49} a + \frac{3}{245}$, $\frac{1}{29871594050379496217316920} a^{20} + \frac{323547026748457193673}{29871594050379496217316920} a^{19} + \frac{116145481820952934435}{1493579702518974810865846} a^{18} - \frac{1373238586499250571185}{1493579702518974810865846} a^{17} + \frac{9342348347108022034213}{2133685289312821158379780} a^{16} - \frac{31137007846704184542267}{2987159405037949621731692} a^{15} - \frac{1460459824689469972514833}{14935797025189748108658460} a^{14} + \frac{177164592504209916879734}{3733949256297437027164615} a^{13} + \frac{1115814694039993140793907}{7467898512594874054329230} a^{12} - \frac{817330437740328048303041}{3733949256297437027164615} a^{11} + \frac{266449703515420625237202}{3733949256297437027164615} a^{10} + \frac{836592844154167196710043}{3733949256297437027164615} a^{9} - \frac{982232779944673931198813}{7467898512594874054329230} a^{8} + \frac{68500486996568158838749}{746789851259487405432923} a^{7} + \frac{3011658329441530159505463}{7467898512594874054329230} a^{6} - \frac{844116823386443938848399}{7467898512594874054329230} a^{5} - \frac{497449288673627106673292}{3733949256297437027164615} a^{4} + \frac{1847251208711794590759856}{3733949256297437027164615} a^{3} - \frac{158973714349864774402157}{746789851259487405432923} a^{2} - \frac{810862068390178876641282}{3733949256297437027164615} a + \frac{8817732586940642158727}{533421322328205289594945}$

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:  \( 713026797535 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_7:C_3$ (as 21T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 63
The 15 conjugacy class representatives for $C_3\times C_7:C_3$
Character table for $C_3\times C_7:C_3$

Intermediate fields

\(\Q(\zeta_{7})^+\), 7.7.525346636864.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $21$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ $21$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{3}$ R ${\href{/LocalNumberField/47.3.0.1}{3} }^{7}$ $21$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
2Data not computed
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$43$$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$