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Properties

Label 21.3.81296592544...1663.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,7^{17}\cdot 83^{18}$
Root discriminant $213.33$
Ramified primes $7, 83$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times D_7$ (as 21T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-581915363, -3420907499, -1381928003, 8934318451, -5051469101, -226790361, 668343838, 38398895, -104698907, 5074902, 12122450, -2200457, -722735, 172662, 35690, -4791, -3326, 1025, -34, 21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 1025*x^17 - 3326*x^16 - 4791*x^15 + 35690*x^14 + 172662*x^13 - 722735*x^12 - 2200457*x^11 + 12122450*x^10 + 5074902*x^9 - 104698907*x^8 + 38398895*x^7 + 668343838*x^6 - 226790361*x^5 - 5051469101*x^4 + 8934318451*x^3 - 1381928003*x^2 - 3420907499*x - 581915363)
 
gp: K = bnfinit(x^21 - 7*x^20 + 21*x^19 - 34*x^18 + 1025*x^17 - 3326*x^16 - 4791*x^15 + 35690*x^14 + 172662*x^13 - 722735*x^12 - 2200457*x^11 + 12122450*x^10 + 5074902*x^9 - 104698907*x^8 + 38398895*x^7 + 668343838*x^6 - 226790361*x^5 - 5051469101*x^4 + 8934318451*x^3 - 1381928003*x^2 - 3420907499*x - 581915363, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 1025 x^{17} - 3326 x^{16} - 4791 x^{15} + 35690 x^{14} + 172662 x^{13} - 722735 x^{12} - 2200457 x^{11} + 12122450 x^{10} + 5074902 x^{9} - 104698907 x^{8} + 38398895 x^{7} + 668343838 x^{6} - 226790361 x^{5} - 5051469101 x^{4} + 8934318451 x^{3} - 1381928003 x^{2} - 3420907499 x - 581915363 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-8129659254492969220523049692132298709650136981663=-\,7^{17}\cdot 83^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $213.33$
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:  $7, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{29} a^{17} + \frac{9}{29} a^{16} - \frac{11}{29} a^{15} + \frac{2}{29} a^{14} - \frac{10}{29} a^{12} - \frac{12}{29} a^{11} - \frac{12}{29} a^{10} - \frac{7}{29} a^{9} - \frac{7}{29} a^{8} - \frac{10}{29} a^{7} - \frac{3}{29} a^{5} - \frac{3}{29} a^{3} - \frac{6}{29} a^{2} + \frac{11}{29} a$, $\frac{1}{2407} a^{18} - \frac{14}{2407} a^{17} + \frac{739}{2407} a^{16} - \frac{383}{2407} a^{15} - \frac{655}{2407} a^{14} + \frac{715}{2407} a^{13} + \frac{711}{2407} a^{12} - \frac{142}{2407} a^{11} - \frac{833}{2407} a^{10} - \frac{687}{2407} a^{9} + \frac{64}{2407} a^{8} + \frac{607}{2407} a^{7} + \frac{432}{2407} a^{6} - \frac{424}{2407} a^{5} + \frac{432}{2407} a^{4} - \frac{198}{2407} a^{3} - \frac{547}{2407} a^{2} - \frac{1123}{2407} a - \frac{2}{83}$, $\frac{1}{2407} a^{19} - \frac{38}{2407} a^{17} - \frac{80}{2407} a^{16} + \frac{374}{2407} a^{15} + \frac{11}{2407} a^{14} + \frac{1093}{2407} a^{13} + \frac{1180}{2407} a^{12} - \frac{663}{2407} a^{11} - \frac{563}{2407} a^{10} - \frac{673}{2407} a^{9} + \frac{756}{2407} a^{8} + \frac{298}{2407} a^{7} + \frac{810}{2407} a^{6} + \frac{1053}{2407} a^{5} + \frac{1036}{2407} a^{4} + \frac{831}{2407} a^{3} - \frac{481}{2407} a^{2} - \frac{508}{2407} a - \frac{28}{83}$, $\frac{1}{10462202803108373688005941486098075082212773249575082798965591122587847384405527976881899058069} a^{20} - \frac{388798106084207171250055046274979403498860534912876209486510600858356432199006188902689613}{10462202803108373688005941486098075082212773249575082798965591122587847384405527976881899058069} a^{19} - \frac{122566829183927638517268494309503870470396410194924195356063044513039048991546329253791220}{804784831008336437538918575853698083247136403813467907612737778660603644954271382837069158313} a^{18} + \frac{88797591181321747855061627605272487141138195545708915516277129664896694296142704005820417015}{10462202803108373688005941486098075082212773249575082798965591122587847384405527976881899058069} a^{17} - \frac{47025838979518755954486813537055548511138896618940844507703100982284753671588837040005939691}{126050636182028598650673993808410543159190039151507021674284230392624667281994312974480711543} a^{16} - \frac{11506615373950666697098346875858921894863309701005494386669435049544107389442267651965136290}{804784831008336437538918575853698083247136403813467907612737778660603644954271382837069158313} a^{15} - \frac{3693384688747026164434882089479934047788928626737672821996006499708214000525944522939297820808}{10462202803108373688005941486098075082212773249575082798965591122587847384405527976881899058069} a^{14} - \frac{5217192676148077605120501477303324484801353187776942117820003795755824207050190431772671138575}{10462202803108373688005941486098075082212773249575082798965591122587847384405527976881899058069} a^{13} - \frac{3169476392806435315257865964468582824774103064976633889641188100615515538752194337332212579723}{10462202803108373688005941486098075082212773249575082798965591122587847384405527976881899058069} a^{12} - \frac{1579808239565317059930245030688438160341855242481476031632008486245477893931039913064877837977}{10462202803108373688005941486098075082212773249575082798965591122587847384405527976881899058069} a^{11} - \frac{1077932519072672374017212565727207737709721342811259712390178322703267336136370122957873768909}{10462202803108373688005941486098075082212773249575082798965591122587847384405527976881899058069} a^{10} - \frac{909843446058025279762996734744005165456751979053740375411556320895851148982775046377704133028}{10462202803108373688005941486098075082212773249575082798965591122587847384405527976881899058069} a^{9} - \frac{2681898083872369168625726953993524677531198991464885412583282844318056102161466406955731303656}{10462202803108373688005941486098075082212773249575082798965591122587847384405527976881899058069} a^{8} + \frac{4757367061538064710524423009786271859992800913481906219226377742465676971313900917797156215802}{10462202803108373688005941486098075082212773249575082798965591122587847384405527976881899058069} a^{7} - \frac{4317382908973296346461925238321625505271776251862105748004663416696197908524993644023390105712}{10462202803108373688005941486098075082212773249575082798965591122587847384405527976881899058069} a^{6} - \frac{3133907651232597098366006202062949001739773974537993502811833505353552999431877078992077264177}{10462202803108373688005941486098075082212773249575082798965591122587847384405527976881899058069} a^{5} + \frac{3442579928270682648368052915649012921704340553102440996061730363309061797610972887031468692130}{10462202803108373688005941486098075082212773249575082798965591122587847384405527976881899058069} a^{4} - \frac{5132634973060588091646416113485391179319268947392853419043512393907066133382880747502743000099}{10462202803108373688005941486098075082212773249575082798965591122587847384405527976881899058069} a^{3} + \frac{1270868533020085858079010678513432820596043280455061344274330429617562392730170243927897126335}{10462202803108373688005941486098075082212773249575082798965591122587847384405527976881899058069} a^{2} - \frac{4799064337210042109752581976854977266959277031284574151557241054128100283548303090547198010640}{10462202803108373688005941486098075082212773249575082798965591122587847384405527976881899058069} a + \frac{19136544088628873173481160354939867094909228110741114788368426314367527674666428419315553690}{360765613900288747862273844348209485593543905157761475826399693882339564979500964720065484761}$

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:  $11$
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:  \( 31367470867428572 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times D_7$ (as 21T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 42
The 15 conjugacy class representatives for $C_3\times D_7$
Character table for $C_3\times D_7$

Intermediate fields

\(\Q(\zeta_{7})^+\), 7.1.112140548065567.1

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

Sibling fields

Galois closure: 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 $21$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R $21$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{21}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ $21$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$83$83.7.6.1$x^{7} - 83$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$
83.7.6.1$x^{7} - 83$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$
83.7.6.1$x^{7} - 83$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$