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Properties

Label 21.21.4003897875...8944.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 199^{14}$
Root discriminant $61.74$
Ramified primes $2, 199$
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![32573, -168018, -57751, 1163183, -581659, -2931914, 2382104, 3271777, -3422708, -1549027, 2184149, 238721, -675808, 26142, 107597, -12180, -8838, 1377, 346, -63, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 5*x^20 - 63*x^19 + 346*x^18 + 1377*x^17 - 8838*x^16 - 12180*x^15 + 107597*x^14 + 26142*x^13 - 675808*x^12 + 238721*x^11 + 2184149*x^10 - 1549027*x^9 - 3422708*x^8 + 3271777*x^7 + 2382104*x^6 - 2931914*x^5 - 581659*x^4 + 1163183*x^3 - 57751*x^2 - 168018*x + 32573)
 
gp: K = bnfinit(x^21 - 5*x^20 - 63*x^19 + 346*x^18 + 1377*x^17 - 8838*x^16 - 12180*x^15 + 107597*x^14 + 26142*x^13 - 675808*x^12 + 238721*x^11 + 2184149*x^10 - 1549027*x^9 - 3422708*x^8 + 3271777*x^7 + 2382104*x^6 - 2931914*x^5 - 581659*x^4 + 1163183*x^3 - 57751*x^2 - 168018*x + 32573, 1)
 

Normalized defining polynomial

\( x^{21} - 5 x^{20} - 63 x^{19} + 346 x^{18} + 1377 x^{17} - 8838 x^{16} - 12180 x^{15} + 107597 x^{14} + 26142 x^{13} - 675808 x^{12} + 238721 x^{11} + 2184149 x^{10} - 1549027 x^{9} - 3422708 x^{8} + 3271777 x^{7} + 2382104 x^{6} - 2931914 x^{5} - 581659 x^{4} + 1163183 x^{3} - 57751 x^{2} - 168018 x + 32573 \)

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:  \(40038978752353387663316403260542418944=2^{18}\cdot 199^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.74$
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:  $2, 199$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{6} a^{18} - \frac{1}{6} a^{17} + \frac{1}{6} a^{15} + \frac{1}{6} a^{13} - \frac{1}{6} a^{12} - \frac{1}{6} a^{11} - \frac{1}{6} a^{10} + \frac{1}{3} a^{9} - \frac{1}{6} a^{8} + \frac{1}{3} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{6} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{942} a^{19} + \frac{9}{314} a^{18} + \frac{16}{471} a^{17} + \frac{116}{471} a^{16} - \frac{107}{942} a^{15} + \frac{85}{942} a^{14} + \frac{22}{157} a^{13} - \frac{173}{942} a^{12} + \frac{212}{471} a^{11} + \frac{179}{471} a^{10} - \frac{181}{471} a^{9} - \frac{239}{942} a^{8} - \frac{317}{942} a^{7} - \frac{139}{314} a^{6} - \frac{232}{471} a^{5} - \frac{191}{471} a^{4} - \frac{46}{157} a^{3} + \frac{78}{157} a^{2} + \frac{28}{157} a - \frac{299}{942}$, $\frac{1}{76330959815693555773593125710767850752500814} a^{20} - \frac{16258561590900340073356581077476149605117}{76330959815693555773593125710767850752500814} a^{19} - \frac{3857956606732653154143474102630626630367979}{76330959815693555773593125710767850752500814} a^{18} + \frac{6058819603648083245005403422756238711280645}{25443653271897851924531041903589283584166938} a^{17} + \frac{3353774836082551306558146094429130549948677}{38165479907846777886796562855383925376250407} a^{16} - \frac{2560965682898077069226101998746710431192137}{38165479907846777886796562855383925376250407} a^{15} - \frac{12343336893043994148168457253166584580528881}{76330959815693555773593125710767850752500814} a^{14} + \frac{2357812671621815176842288046595300592300668}{38165479907846777886796562855383925376250407} a^{13} + \frac{16861516073778377737005465543225980988133061}{76330959815693555773593125710767850752500814} a^{12} - \frac{24337672123389951416049004983252774901152547}{76330959815693555773593125710767850752500814} a^{11} - \frac{460899465753298650116837677043954841873188}{38165479907846777886796562855383925376250407} a^{10} + \frac{28166235474120992321331431346551974195468085}{76330959815693555773593125710767850752500814} a^{9} + \frac{21990713831980150225170803828988625570967555}{76330959815693555773593125710767850752500814} a^{8} + \frac{33993562160597689861219923416730911565375421}{76330959815693555773593125710767850752500814} a^{7} - \frac{13060227348952522251275870267828946146537759}{76330959815693555773593125710767850752500814} a^{6} - \frac{10876494973189878023243668582098683861832605}{25443653271897851924531041903589283584166938} a^{5} - \frac{26335598729758317519941725799117456066806915}{76330959815693555773593125710767850752500814} a^{4} - \frac{718422974596760574415879647778615511252267}{25443653271897851924531041903589283584166938} a^{3} - \frac{583639765269134260239205958645423037201575}{12721826635948925962265520951794641792083469} a^{2} + \frac{25496780491916067029224361095945191868268899}{76330959815693555773593125710767850752500814} a + \frac{10783699333090150880835003363578249164396949}{38165479907846777886796562855383925376250407}$

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:  \( 1272666465930 \) (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

3.3.39601.1, 7.7.100367308864.1 x7

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

Sibling fields

Degree 7 sibling: 7.7.100367308864.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/5.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.7.0.1}{7} }^{3}$

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
2Data not computed
$199$199.3.2.1$x^{3} - 199$$3$$1$$2$$C_3$$[\ ]_{3}$
199.3.2.1$x^{3} - 199$$3$$1$$2$$C_3$$[\ ]_{3}$
199.3.2.1$x^{3} - 199$$3$$1$$2$$C_3$$[\ ]_{3}$
199.3.2.1$x^{3} - 199$$3$$1$$2$$C_3$$[\ ]_{3}$
199.3.2.1$x^{3} - 199$$3$$1$$2$$C_3$$[\ ]_{3}$
199.3.2.1$x^{3} - 199$$3$$1$$2$$C_3$$[\ ]_{3}$
199.3.2.1$x^{3} - 199$$3$$1$$2$$C_3$$[\ ]_{3}$