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Properties

Label 21.21.2324348490...6361.1
Degree $21$
Signature $[21, 0]$
Discriminant $7^{32}\cdot 29^{18}$
Root discriminant $347.74$
Ramified primes $7, 29$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $C_7:(C_7:C_3)$ (as 21T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10374903606295432, 2797998936825036, -10218999224214690, -5168916513642035, 1409517134374440, 1032834108286134, -72906058674540, -88767002725137, 1795715730900, 4167927010247, -22704451770, -117590417217, 142347660, 2079406140, -350610, -23218328, 0, 158949, 0, -609, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 609*x^19 + 158949*x^17 - 23218328*x^15 - 350610*x^14 + 2079406140*x^13 + 142347660*x^12 - 117590417217*x^11 - 22704451770*x^10 + 4167927010247*x^9 + 1795715730900*x^8 - 88767002725137*x^7 - 72906058674540*x^6 + 1032834108286134*x^5 + 1409517134374440*x^4 - 5168916513642035*x^3 - 10218999224214690*x^2 + 2797998936825036*x + 10374903606295432)
 
gp: K = bnfinit(x^21 - 609*x^19 + 158949*x^17 - 23218328*x^15 - 350610*x^14 + 2079406140*x^13 + 142347660*x^12 - 117590417217*x^11 - 22704451770*x^10 + 4167927010247*x^9 + 1795715730900*x^8 - 88767002725137*x^7 - 72906058674540*x^6 + 1032834108286134*x^5 + 1409517134374440*x^4 - 5168916513642035*x^3 - 10218999224214690*x^2 + 2797998936825036*x + 10374903606295432, 1)
 

Normalized defining polynomial

\( x^{21} - 609 x^{19} + 158949 x^{17} - 23218328 x^{15} - 350610 x^{14} + 2079406140 x^{13} + 142347660 x^{12} - 117590417217 x^{11} - 22704451770 x^{10} + 4167927010247 x^{9} + 1795715730900 x^{8} - 88767002725137 x^{7} - 72906058674540 x^{6} + 1032834108286134 x^{5} + 1409517134374440 x^{4} - 5168916513642035 x^{3} - 10218999224214690 x^{2} + 2797998936825036 x + 10374903606295432 \)

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:  \(232434849099118488288057964926900122812572027739476361=7^{32}\cdot 29^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $347.74$
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:  $7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2436} a^{7} - \frac{1}{12} a^{5} - \frac{1}{6} a^{3} - \frac{1}{12} a + \frac{1}{42}$, $\frac{1}{2436} a^{8} - \frac{1}{12} a^{6} - \frac{1}{6} a^{4} - \frac{1}{12} a^{2} + \frac{1}{42} a$, $\frac{1}{2436} a^{9} - \frac{1}{12} a^{5} + \frac{1}{12} a^{3} + \frac{1}{42} a^{2} + \frac{1}{12} a - \frac{1}{6}$, $\frac{1}{70644} a^{10} + \frac{1}{12} a^{6} - \frac{1}{12} a^{4} + \frac{95}{609} a^{3} - \frac{1}{12} a^{2} + \frac{1}{6} a$, $\frac{1}{282576} a^{11} - \frac{1}{282576} a^{10} + \frac{1}{9744} a^{9} - \frac{1}{4872} a^{8} + \frac{7}{48} a^{6} + \frac{3}{16} a^{5} - \frac{347}{3248} a^{4} + \frac{681}{1624} a^{3} + \frac{79}{336} a^{2} - \frac{5}{28} a - \frac{1}{4}$, $\frac{1}{57362928} a^{12} + \frac{1}{659344} a^{11} + \frac{5}{1978032} a^{10} + \frac{5}{34104} a^{9} - \frac{1}{17052} a^{8} - \frac{1}{22736} a^{7} - \frac{43}{336} a^{6} + \frac{190591}{1978032} a^{5} - \frac{13}{11368} a^{4} - \frac{109}{68208} a^{3} - \frac{37}{588} a^{2} - \frac{107}{588} a + \frac{67}{147}$, $\frac{1}{57362928} a^{13} + \frac{1}{659344} a^{11} - \frac{1}{164836} a^{10} + \frac{1}{68208} a^{9} - \frac{5}{68208} a^{8} - \frac{1}{5684} a^{7} - \frac{409883}{1978032} a^{6} + \frac{3}{56} a^{5} - \frac{739}{34104} a^{4} - \frac{15319}{68208} a^{3} - \frac{355}{2352} a^{2} - \frac{26}{147} a - \frac{223}{588}$, $\frac{1}{144726667344} a^{14} - \frac{1}{356469624} a^{12} + \frac{11}{24584112} a^{10} - \frac{5}{141288} a^{8} - \frac{215375}{1247643684} a^{7} + \frac{1}{696} a^{6} + \frac{215375}{6146028} a^{5} - \frac{1}{36} a^{4} - \frac{3443}{105966} a^{3} + \frac{29}{144} a^{2} + \frac{3443}{7308} a + \frac{7645}{51156}$, $\frac{1}{289453334688} a^{15} - \frac{1}{712939248} a^{13} + \frac{11}{49168224} a^{11} - \frac{5}{282576} a^{9} + \frac{148397}{1247643684} a^{8} - \frac{1}{9744} a^{7} + \frac{694055}{3073014} a^{6} + \frac{11}{72} a^{5} - \frac{5276}{52983} a^{4} + \frac{125}{288} a^{3} - \frac{205}{3654} a^{2} - \frac{50819}{102312} a + \frac{19}{42}$, $\frac{1}{289453334688} a^{16} + \frac{1}{344177568} a^{12} - \frac{1}{659344} a^{11} - \frac{1}{5934096} a^{10} + \frac{53441}{712939248} a^{9} - \frac{1}{22736} a^{8} - \frac{5743}{43022196} a^{7} - \frac{179}{1008} a^{6} - \frac{487153}{1978032} a^{5} - \frac{6637}{409248} a^{4} + \frac{2599}{34104} a^{3} + \frac{48593}{102312} a^{2} - \frac{100}{441} a - \frac{215}{441}$, $\frac{1}{58759026941664} a^{17} - \frac{1}{2026173342816} a^{16} - \frac{1}{1013086671408} a^{15} - \frac{1}{337695557136} a^{14} - \frac{13}{9981149472} a^{13} - \frac{17}{9981149472} a^{12} - \frac{41}{57362928} a^{11} - \frac{5063819}{1013086671408} a^{10} - \frac{254363}{8733505788} a^{9} + \frac{1961263}{34934023152} a^{8} - \frac{5065961}{34934023152} a^{7} - \frac{868444}{10755549} a^{6} + \frac{20811491}{344177568} a^{5} + \frac{1275895}{11868192} a^{4} - \frac{15435775}{41538672} a^{3} + \frac{620177}{1432368} a^{2} - \frac{157165}{358092} a + \frac{119977}{358092}$, $\frac{1}{1527734700483264} a^{18} + \frac{1}{169748300053696} a^{17} + \frac{43}{26340253456608} a^{16} - \frac{25}{17560168971072} a^{15} + \frac{19}{5853389657024} a^{14} - \frac{1}{86503295424} a^{13} - \frac{31}{9981149472} a^{12} - \frac{40121903}{52680506913216} a^{11} + \frac{6366291}{1463347414256} a^{10} - \frac{18509495}{151380766992} a^{9} + \frac{40069919}{227071150488} a^{8} - \frac{107136401}{908284601952} a^{7} + \frac{21221065}{2982872256} a^{6} + \frac{191586827}{8948616768} a^{5} - \frac{14604137}{180000912} a^{4} - \frac{24190585}{74483136} a^{3} + \frac{11956067}{37241568} a^{2} + \frac{455227}{6206928} a - \frac{601933}{3103464}$, $\frac{1}{44304306314014656} a^{19} + \frac{1}{218247814354752} a^{17} + \frac{1}{17560168971072} a^{16} - \frac{41}{26340253456608} a^{15} - \frac{1}{365836853564} a^{14} - \frac{23}{86503295424} a^{13} + \frac{4419139}{1527734700483264} a^{12} - \frac{209}{229451712} a^{11} + \frac{41051}{1881446675472} a^{10} + \frac{967301}{113535575244} a^{9} - \frac{19387889}{908284601952} a^{8} - \frac{48054505}{1816569203904} a^{7} - \frac{9975593}{1118577096} a^{6} - \frac{6967644407}{62640317376} a^{5} - \frac{15776273}{102857664} a^{4} + \frac{3092675}{7912128} a^{3} - \frac{5447405}{12413856} a^{2} + \frac{136811}{6206928} a - \frac{1541753}{3103464}$, $\frac{1}{44304306314014656} a^{20} - \frac{1}{763867350241632} a^{17} + \frac{5}{3762893350944} a^{16} + \frac{23}{17560168971072} a^{15} - \frac{31}{13170126728304} a^{14} + \frac{2704577}{381933675120816} a^{13} + \frac{19}{6654099648} a^{12} + \frac{8055623}{7525786701888} a^{11} + \frac{21490993}{6585063364152} a^{10} + \frac{1643331}{14417215904} a^{9} + \frac{77804555}{1816569203904} a^{8} + \frac{183113963}{908284601952} a^{7} - \frac{4532574359}{31320158688} a^{6} + \frac{36590899}{1491436128} a^{5} + \frac{3773963}{308572992} a^{4} + \frac{885256217}{2160010944} a^{3} - \frac{125689}{5320224} a^{2} + \frac{128675}{1432368} a + \frac{160913}{1034488}$

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

Class group and class number

$C_{7}$, which has order $7$ (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:  \( 80111690644900000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_7:(C_7:C_3)$ (as 21T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 147
The 19 conjugacy class representatives for $C_7:(C_7:C_3)$
Character table for $C_7:(C_7:C_3)$

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 sibling: 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
7Data not computed
$29$29.7.6.3$x^{7} + 928$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.3$x^{7} + 928$$7$$1$$6$$C_7$$[\ ]_{7}$
29.7.6.3$x^{7} + 928$$7$$1$$6$$C_7$$[\ ]_{7}$