↑

Properties

Label 21.3.17184371679...0183.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,7^{17}\cdot 127^{18}$
Root discriminant $307.18$
Ramified primes $7, 127$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $F_7$ (as 21T4)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-322494640909, 175769815592, 28879725259, -95684899751, 9054965094, 3887930312, -6303984553, 105257496, 399984668, -37794391, 23235788, 389109, -3477004, 37681, -14067, -12259, 4693, -860, -34, 21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 21*x^19 - 34*x^18 - 860*x^17 + 4693*x^16 - 12259*x^15 - 14067*x^14 + 37681*x^13 - 3477004*x^12 + 389109*x^11 + 23235788*x^10 - 37794391*x^9 + 399984668*x^8 + 105257496*x^7 - 6303984553*x^6 + 3887930312*x^5 + 9054965094*x^4 - 95684899751*x^3 + 28879725259*x^2 + 175769815592*x - 322494640909)
 
gp: K = bnfinit(x^21 - 7*x^20 + 21*x^19 - 34*x^18 - 860*x^17 + 4693*x^16 - 12259*x^15 - 14067*x^14 + 37681*x^13 - 3477004*x^12 + 389109*x^11 + 23235788*x^10 - 37794391*x^9 + 399984668*x^8 + 105257496*x^7 - 6303984553*x^6 + 3887930312*x^5 + 9054965094*x^4 - 95684899751*x^3 + 28879725259*x^2 + 175769815592*x - 322494640909, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} - 860 x^{17} + 4693 x^{16} - 12259 x^{15} - 14067 x^{14} + 37681 x^{13} - 3477004 x^{12} + 389109 x^{11} + 23235788 x^{10} - 37794391 x^{9} + 399984668 x^{8} + 105257496 x^{7} - 6303984553 x^{6} + 3887930312 x^{5} + 9054965094 x^{4} - 95684899751 x^{3} + 28879725259 x^{2} + 175769815592 x - 322494640909 \)

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:  \(-17184371679714375406342276014583508152741413197410183=-\,7^{17}\cdot 127^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $307.18$
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:  $7, 127$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{14} - \frac{1}{4}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{12} - \frac{1}{4} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} + \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{3}{8} a^{4} - \frac{1}{2} a^{3} - \frac{3}{8} a^{2} - \frac{3}{8} a + \frac{3}{8}$, $\frac{1}{8} a^{16} - \frac{1}{8} a^{13} - \frac{1}{8} a^{11} + \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} + \frac{1}{4} a^{4} + \frac{1}{8} a^{3} + \frac{3}{8} a^{2} + \frac{3}{8} a - \frac{1}{4}$, $\frac{1}{104} a^{17} - \frac{1}{104} a^{16} + \frac{1}{52} a^{15} - \frac{3}{104} a^{14} + \frac{1}{104} a^{13} - \frac{9}{104} a^{12} + \frac{21}{104} a^{11} - \frac{15}{104} a^{10} - \frac{19}{104} a^{9} + \frac{5}{104} a^{8} + \frac{19}{104} a^{7} + \frac{15}{104} a^{6} - \frac{27}{104} a^{5} - \frac{25}{104} a^{4} + \frac{15}{52} a^{3} - \frac{7}{26} a^{2} + \frac{25}{104} a + \frac{3}{26}$, $\frac{1}{13208} a^{18} - \frac{45}{13208} a^{17} + \frac{657}{13208} a^{16} + \frac{13}{508} a^{15} - \frac{23}{13208} a^{14} - \frac{787}{6604} a^{13} - \frac{107}{3302} a^{12} + \frac{447}{6604} a^{11} + \frac{280}{1651} a^{10} + \frac{519}{3302} a^{9} - \frac{159}{6604} a^{8} - \frac{1535}{6604} a^{7} - \frac{1559}{6604} a^{6} - \frac{1045}{3302} a^{5} - \frac{2575}{13208} a^{4} + \frac{2357}{13208} a^{3} + \frac{5183}{13208} a^{2} + \frac{1171}{3302} a - \frac{1503}{13208}$, $\frac{1}{66040} a^{19} + \frac{1}{33020} a^{18} - \frac{63}{13208} a^{17} + \frac{1829}{33020} a^{16} - \frac{1663}{66040} a^{15} + \frac{137}{1270} a^{14} - \frac{3921}{66040} a^{13} + \frac{6813}{66040} a^{12} + \frac{12127}{66040} a^{11} + \frac{1057}{66040} a^{10} + \frac{12799}{66040} a^{9} - \frac{8999}{66040} a^{8} + \frac{3087}{66040} a^{7} - \frac{4803}{66040} a^{6} - \frac{14237}{33020} a^{5} + \frac{27763}{66040} a^{4} + \frac{1487}{6604} a^{3} + \frac{17}{40} a^{2} - \frac{6819}{33020} a - \frac{791}{66040}$, $\frac{1}{246451546250762088127673247635059403113057823987048324830440439813464720522651764878513425315057800} a^{20} + \frac{202679536061278466485391147546518861706906674024144095119885996290468141946632220825577894941}{49290309250152417625534649527011880622611564797409664966088087962692944104530352975702685063011560} a^{19} + \frac{1054987798043170162122291203866066332595959334458560568679721346884335210496092518121707549332}{30806443281345261015959155954382425389132227998381040603805054976683090065331470609814178164382225} a^{18} - \frac{9431547395687893062294575309990244350896132561239163882002528964062697025442512623971946865231}{123225773125381044063836623817529701556528911993524162415220219906732360261325882439256712657528900} a^{17} + \frac{8375081493566269574183654433875161882743564494765553864424274031013345379593358055923795993946171}{246451546250762088127673247635059403113057823987048324830440439813464720522651764878513425315057800} a^{16} + \frac{513652426760024506417305641741686968883469797969181258712540929061266367383141508425426753096959}{49290309250152417625534649527011880622611564797409664966088087962692944104530352975702685063011560} a^{15} + \frac{138733574500419977099813111805319068139125193917942073015291304514253316461762395398180487684439}{61612886562690522031918311908764850778264455996762081207610109953366180130662941219628356328764450} a^{14} + \frac{3234059444664379622889653464076980538498856838552834542287500826378831898710298539988821137858071}{49290309250152417625534649527011880622611564797409664966088087962692944104530352975702685063011560} a^{13} - \frac{6857788430462685245964674211943848080057489526392201514351882314838386221269246283621924428961821}{61612886562690522031918311908764850778264455996762081207610109953366180130662941219628356328764450} a^{12} - \frac{59124403946738604712159750734928787885757430608732565536589735025780473947550776257603045432356037}{246451546250762088127673247635059403113057823987048324830440439813464720522651764878513425315057800} a^{11} - \frac{2840548535474976771496052259343145226297840579546638530644940285589316774842164212884458447358393}{12322577312538104406383662381752970155652891199352416241522021990673236026132588243925671265752890} a^{10} - \frac{11346478189496113242649859294979492452224703328591711890565528625388239234058187256719185530876657}{246451546250762088127673247635059403113057823987048324830440439813464720522651764878513425315057800} a^{9} - \frac{206483692193973556644295729548896234433242918026121894588125109486114415950752948190071706956545}{1232257731253810440638366238175297015565289119935241624152202199067323602613258824392567126575289} a^{8} - \frac{24128843371286526418894723113603647268094068801808682164509133195646180388025643277814988883470257}{246451546250762088127673247635059403113057823987048324830440439813464720522651764878513425315057800} a^{7} - \frac{14587697266711816093322820736971531353420731073575560050084005120861338866596901758758117194380113}{246451546250762088127673247635059403113057823987048324830440439813464720522651764878513425315057800} a^{6} + \frac{7292685436256435745837697990598797979876094707117646057246748628849621191588319478835315283684583}{123225773125381044063836623817529701556528911993524162415220219906732360261325882439256712657528900} a^{5} + \frac{3223343574145451733159781875860923470937527997611502396168367174712953847519987244913173561882513}{30806443281345261015959155954382425389132227998381040603805054976683090065331470609814178164382225} a^{4} - \frac{116052999536203699486711593287708132488151051664414324726351279989089109509957778486952324156978283}{246451546250762088127673247635059403113057823987048324830440439813464720522651764878513425315057800} a^{3} + \frac{41841487490933508576978330870625593048197044574979157054681387039514801976312384468040021001958153}{246451546250762088127673247635059403113057823987048324830440439813464720522651764878513425315057800} a^{2} + \frac{622471766296341277279791681486632476069183134430627117811645151260331171061866408384543593708}{6763214770877115481000912393936866166659106036966199913019770576659295294254988059234726270995} a + \frac{13123181743612261273178326367544474836392384374743296192712542536579253133895151510264238645960104}{30806443281345261015959155954382425389132227998381040603805054976683090065331470609814178164382225}$

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:  $11$
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:  \( 268187551415576800 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_7$ (as 21T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 42
The 7 conjugacy class representatives for $F_7$
Character table for $F_7$

Intermediate fields

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

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

Sibling fields

Galois closure: data not computed
Degree 7 sibling: data not computed
Degree 14 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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}$
$127$127.7.6.1$x^{7} - 127$$7$$1$$6$$C_7$$[\ ]_{7}$
127.7.6.1$x^{7} - 127$$7$$1$$6$$C_7$$[\ ]_{7}$
127.7.6.1$x^{7} - 127$$7$$1$$6$$C_7$$[\ ]_{7}$