↑

Properties

Label 21.21.2049367385...6801.1
Degree $21$
Signature $[21, 0]$
Discriminant $463^{20}$
Root discriminant $345.66$
Ramified prime $463$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{21}$ (as 21T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1888668673, 23424596481, 84397161408, 119457363636, 42961970730, -53493754115, -49217032511, -1212682637, 11114459489, 2403675074, -1090202912, -359999966, 53278874, 24832342, -1261590, -933778, 10669, 19731, 59, -220, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - x^20 - 220*x^19 + 59*x^18 + 19731*x^17 + 10669*x^16 - 933778*x^15 - 1261590*x^14 + 24832342*x^13 + 53278874*x^12 - 359999966*x^11 - 1090202912*x^10 + 2403675074*x^9 + 11114459489*x^8 - 1212682637*x^7 - 49217032511*x^6 - 53493754115*x^5 + 42961970730*x^4 + 119457363636*x^3 + 84397161408*x^2 + 23424596481*x + 1888668673)
 
gp: K = bnfinit(x^21 - x^20 - 220*x^19 + 59*x^18 + 19731*x^17 + 10669*x^16 - 933778*x^15 - 1261590*x^14 + 24832342*x^13 + 53278874*x^12 - 359999966*x^11 - 1090202912*x^10 + 2403675074*x^9 + 11114459489*x^8 - 1212682637*x^7 - 49217032511*x^6 - 53493754115*x^5 + 42961970730*x^4 + 119457363636*x^3 + 84397161408*x^2 + 23424596481*x + 1888668673, 1)
 

Normalized defining polynomial

\( x^{21} - x^{20} - 220 x^{19} + 59 x^{18} + 19731 x^{17} + 10669 x^{16} - 933778 x^{15} - 1261590 x^{14} + 24832342 x^{13} + 53278874 x^{12} - 359999966 x^{11} - 1090202912 x^{10} + 2403675074 x^{9} + 11114459489 x^{8} - 1212682637 x^{7} - 49217032511 x^{6} - 53493754115 x^{5} + 42961970730 x^{4} + 119457363636 x^{3} + 84397161408 x^{2} + 23424596481 x + 1888668673 \)

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:  \(204936738540452503064519918459921419848199261587556801=463^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $345.66$
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:  $463$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(463\)
Dirichlet character group:    $\lbrace$$\chi_{463}(1,·)$, $\chi_{463}(450,·)$, $\chi_{463}(196,·)$, $\chi_{463}(182,·)$, $\chi_{463}(449,·)$, $\chi_{463}(200,·)$, $\chi_{463}(21,·)$, $\chi_{463}(412,·)$, $\chi_{463}(286,·)$, $\chi_{463}(33,·)$, $\chi_{463}(34,·)$, $\chi_{463}(163,·)$, $\chi_{463}(230,·)$, $\chi_{463}(169,·)$, $\chi_{463}(178,·)$, $\chi_{463}(308,·)$, $\chi_{463}(318,·)$, $\chi_{463}(118,·)$, $\chi_{463}(441,·)$, $\chi_{463}(251,·)$, $\chi_{463}(190,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{101} a^{19} - \frac{32}{101} a^{18} + \frac{18}{101} a^{16} - \frac{17}{101} a^{15} + \frac{27}{101} a^{14} + \frac{33}{101} a^{13} - \frac{50}{101} a^{12} - \frac{12}{101} a^{11} + \frac{47}{101} a^{10} + \frac{20}{101} a^{9} + \frac{38}{101} a^{8} - \frac{43}{101} a^{7} + \frac{10}{101} a^{6} - \frac{18}{101} a^{5} + \frac{50}{101} a^{4} + \frac{39}{101} a^{3} - \frac{32}{101} a^{2} + \frac{34}{101} a - \frac{36}{101}$, $\frac{1}{21177386320080591158077402682952589002934166599918449484499169991275320085740063238713698038607} a^{20} + \frac{86965159837721340256688190222676610457906247995267899420723579690786734659176205185688609285}{21177386320080591158077402682952589002934166599918449484499169991275320085740063238713698038607} a^{19} + \frac{10293871796545219938658114816762835859688168457429344083645943379186861106607617504748394472492}{21177386320080591158077402682952589002934166599918449484499169991275320085740063238713698038607} a^{18} + \frac{4852457073798592942469886955390107311787726188176458929022093502456613187967390965786554516112}{21177386320080591158077402682952589002934166599918449484499169991275320085740063238713698038607} a^{17} - \frac{7743920069241201775291090804889859695898161737553091662589770971428338874915948723109549619288}{21177386320080591158077402682952589002934166599918449484499169991275320085740063238713698038607} a^{16} + \frac{1251355011207425712133964182792007185667827687241962679394426862331995769253389415522710962605}{21177386320080591158077402682952589002934166599918449484499169991275320085740063238713698038607} a^{15} - \frac{8553062775570283292374566793828285381243282406027455582824939838963581631094279620469005543239}{21177386320080591158077402682952589002934166599918449484499169991275320085740063238713698038607} a^{14} - \frac{5924486590003717926748093748885894948732527398193555404432974014811183928466326607419077161889}{21177386320080591158077402682952589002934166599918449484499169991275320085740063238713698038607} a^{13} - \frac{7507371508624380123455294141914034695577164459761929359536062798649310685782374124688307714336}{21177386320080591158077402682952589002934166599918449484499169991275320085740063238713698038607} a^{12} - \frac{2347907397904412347370050406788627148372516440475001139146093399023655465807658691015522782822}{21177386320080591158077402682952589002934166599918449484499169991275320085740063238713698038607} a^{11} + \frac{932427464464118927600225705220941438099464681647381652226801983874635975703185408418730524813}{21177386320080591158077402682952589002934166599918449484499169991275320085740063238713698038607} a^{10} + \frac{9948851001388313421512826416094534242310757675929276745634226862832344458084205386096394627464}{21177386320080591158077402682952589002934166599918449484499169991275320085740063238713698038607} a^{9} + \frac{5570957262509547736360911012725472014386426134946830365631206271413011362564420713154753952959}{21177386320080591158077402682952589002934166599918449484499169991275320085740063238713698038607} a^{8} - \frac{4258558905395141260057946224383646087401980853028728156861614353804476108083665218040032403352}{21177386320080591158077402682952589002934166599918449484499169991275320085740063238713698038607} a^{7} + \frac{1916615080601160749134064595694055843678264326706064702932470395925086246780578221564272519734}{21177386320080591158077402682952589002934166599918449484499169991275320085740063238713698038607} a^{6} + \frac{2657731835728214061592102869469601730838172468015170378165910930259276704465318588964810070458}{21177386320080591158077402682952589002934166599918449484499169991275320085740063238713698038607} a^{5} - \frac{8557039566378774554106305230365077474976969759274037763899103458114245506733104194118352364060}{21177386320080591158077402682952589002934166599918449484499169991275320085740063238713698038607} a^{4} + \frac{5128540859466854331552081520077205676258154184998493735041188744342152034353113569850018588180}{21177386320080591158077402682952589002934166599918449484499169991275320085740063238713698038607} a^{3} - \frac{9973415995271795842826839842080829388107590267807264227409200283598734196782352543889187643170}{21177386320080591158077402682952589002934166599918449484499169991275320085740063238713698038607} a^{2} + \frac{1485592505484104547506979596817623491434530916607015545880595768070413792944077332411378519114}{21177386320080591158077402682952589002934166599918449484499169991275320085740063238713698038607} a - \frac{6481281765616522844967013867673969819470488417057368889761925874840895913231503016345239272239}{21177386320080591158077402682952589002934166599918449484499169991275320085740063238713698038607}$

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:  \( 62050086594131485000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{21}$ (as 21T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 21
The 21 conjugacy class representatives for $C_{21}$
Character table for $C_{21}$ is not computed

Intermediate fields

3.3.214369.1, 7.7.9851127637605409.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $21$ $21$ $21$ ${\href{/LocalNumberField/7.7.0.1}{7} }^{3}$ $21$ $21$ $21$ $21$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{3}$ $21$ $21$ $21$ $21$ $21$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ $21$ $21$

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
463Data not computed