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Properties

Label 21.21.7812869458...2736.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 13^{14}\cdot 31^{14}$
Root discriminant $98.83$
Ramified primes $2, 13, 31$
Class number $3$ (GRH)
Class group $[3]$ (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![-325961, 1883508, 1847525, -19620833, -2336541, 63845912, 12806508, -75363341, -27520996, 32672825, 14463151, -6541447, -3106176, 762604, 340921, -58666, -20148, 2969, 600, -85, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - 85*x^19 + 600*x^18 + 2969*x^17 - 20148*x^16 - 58666*x^15 + 340921*x^14 + 762604*x^13 - 3106176*x^12 - 6541447*x^11 + 14463151*x^10 + 32672825*x^9 - 27520996*x^8 - 75363341*x^7 + 12806508*x^6 + 63845912*x^5 - 2336541*x^4 - 19620833*x^3 + 1847525*x^2 + 1883508*x - 325961)
 
gp: K = bnfinit(x^21 - 7*x^20 - 85*x^19 + 600*x^18 + 2969*x^17 - 20148*x^16 - 58666*x^15 + 340921*x^14 + 762604*x^13 - 3106176*x^12 - 6541447*x^11 + 14463151*x^10 + 32672825*x^9 - 27520996*x^8 - 75363341*x^7 + 12806508*x^6 + 63845912*x^5 - 2336541*x^4 - 19620833*x^3 + 1847525*x^2 + 1883508*x - 325961, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} - 85 x^{19} + 600 x^{18} + 2969 x^{17} - 20148 x^{16} - 58666 x^{15} + 340921 x^{14} + 762604 x^{13} - 3106176 x^{12} - 6541447 x^{11} + 14463151 x^{10} + 32672825 x^{9} - 27520996 x^{8} - 75363341 x^{7} + 12806508 x^{6} + 63845912 x^{5} - 2336541 x^{4} - 19620833 x^{3} + 1847525 x^{2} + 1883508 x - 325961 \)

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:  \(781286945806524287560821363867897500532736=2^{18}\cdot 13^{14}\cdot 31^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $98.83$
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:  $2, 13, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{71} a^{16} - \frac{6}{71} a^{15} + \frac{4}{71} a^{14} - \frac{20}{71} a^{13} + \frac{19}{71} a^{12} - \frac{18}{71} a^{11} - \frac{27}{71} a^{10} + \frac{20}{71} a^{9} + \frac{8}{71} a^{8} - \frac{4}{71} a^{7} + \frac{6}{71} a^{6} + \frac{34}{71} a^{5} - \frac{9}{71} a^{4} - \frac{33}{71} a^{3} + \frac{14}{71} a^{2} - \frac{17}{71} a$, $\frac{1}{71} a^{17} - \frac{32}{71} a^{15} + \frac{4}{71} a^{14} - \frac{30}{71} a^{13} + \frac{25}{71} a^{12} + \frac{7}{71} a^{11} - \frac{14}{71} a^{9} - \frac{27}{71} a^{8} - \frac{18}{71} a^{7} - \frac{1}{71} a^{6} - \frac{18}{71} a^{5} - \frac{16}{71} a^{4} + \frac{29}{71} a^{3} - \frac{4}{71} a^{2} - \frac{31}{71} a$, $\frac{1}{4615} a^{18} - \frac{28}{4615} a^{17} - \frac{8}{4615} a^{16} + \frac{472}{4615} a^{15} + \frac{522}{4615} a^{14} - \frac{349}{923} a^{13} - \frac{521}{4615} a^{12} + \frac{1644}{4615} a^{11} + \frac{2036}{4615} a^{10} + \frac{136}{355} a^{9} + \frac{6}{355} a^{8} - \frac{138}{355} a^{7} - \frac{556}{4615} a^{6} + \frac{97}{4615} a^{5} - \frac{2011}{4615} a^{4} - \frac{2034}{4615} a^{3} - \frac{1571}{4615} a^{2} - \frac{37}{4615} a - \frac{29}{65}$, $\frac{1}{3604315} a^{19} + \frac{1}{55451} a^{18} + \frac{6423}{3604315} a^{17} + \frac{17213}{3604315} a^{16} + \frac{873363}{3604315} a^{15} - \frac{756274}{3604315} a^{14} - \frac{22086}{50765} a^{13} - \frac{319419}{3604315} a^{12} - \frac{64252}{3604315} a^{11} - \frac{820934}{3604315} a^{10} - \frac{47566}{277255} a^{9} + \frac{2289}{5041} a^{8} - \frac{215368}{3604315} a^{7} - \frac{75726}{3604315} a^{6} + \frac{270528}{720863} a^{5} + \frac{778143}{3604315} a^{4} - \frac{10733}{327665} a^{3} + \frac{246021}{720863} a^{2} + \frac{123414}{720863} a - \frac{2957}{50765}$, $\frac{1}{227242246415669696494986175501952398559147580120441695} a^{20} + \frac{1734377081467157643837347344057902903485523952}{45448449283133939298997235100390479711829516024088339} a^{19} - \frac{217208120081882621281060149672097721815141144859}{227242246415669696494986175501952398559147580120441695} a^{18} - \frac{112506848079709749824556559083436719343755493511332}{17480172801205361268845090423227107581472890778495515} a^{17} + \frac{852687829175134590229404175738824270489861167839174}{227242246415669696494986175501952398559147580120441695} a^{16} - \frac{7302720569159000694469597544119382670169925894684158}{227242246415669696494986175501952398559147580120441695} a^{15} - \frac{21048868634646484755429139974745609023763058535298495}{45448449283133939298997235100390479711829516024088339} a^{14} + \frac{30236037141124488450946122407686519639835664768351591}{227242246415669696494986175501952398559147580120441695} a^{13} + \frac{8375399873562417131096027913002285511319035956887227}{45448449283133939298997235100390479711829516024088339} a^{12} + \frac{80022567909976808106646802691622635929602262145044578}{227242246415669696494986175501952398559147580120441695} a^{11} - \frac{877998066498654423574750231488561140640074535431956}{45448449283133939298997235100390479711829516024088339} a^{10} + \frac{7058887709233946732098129780642566699040796674190043}{17480172801205361268845090423227107581472890778495515} a^{9} + \frac{23749418947546735005404416243607712160228856203576806}{227242246415669696494986175501952398559147580120441695} a^{8} + \frac{46893163358584782894549159377686574422281606215725757}{227242246415669696494986175501952398559147580120441695} a^{7} - \frac{87113863171067357077330568740131705877761726381097993}{227242246415669696494986175501952398559147580120441695} a^{6} - \frac{82625668827412446284079636656015394780404809341533816}{227242246415669696494986175501952398559147580120441695} a^{5} + \frac{110057449687680405003040656813760577734007474044717569}{227242246415669696494986175501952398559147580120441695} a^{4} + \frac{90688670613944632448463015569337887364287488441846873}{227242246415669696494986175501952398559147580120441695} a^{3} + \frac{77491979681428642770275452280046635170484827032773167}{227242246415669696494986175501952398559147580120441695} a^{2} - \frac{67616061548827258277343872960805213582383046358574233}{227242246415669696494986175501952398559147580120441695} a - \frac{24912669455209361674596359422251971109179783809082}{3200595019939009809788537683126090120551374367893545}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 107348023084000 \) (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

3.3.162409.2, 7.7.1688107729984.1 x7

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

Sibling fields

Degree 7 sibling: 7.7.1688107729984.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/37.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
2Data not computed
$13$13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
$31$31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$