↑

Properties

Label 21.1.38529903145...0000.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{33}\cdot 3^{19}\cdot 5^{19}\cdot 7^{21}\cdot 167^{7}$
Root discriminant $1327.80$
Ramified primes $2, 3, 5, 7, 167$
Class number Not computed
Class group Not computed
Galois group $S_3\times F_7$ (as 21T15)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![19907441280, -60906867840, 39186141120, 4102539840, 18597509280, -10704143520, 2861475120, -2671533936, 867775608, -9581544, -12561948, -3354876, 2834454, -763602, -28459, 51877, -7371, -1099, 427, -21, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - 21*x^19 + 427*x^18 - 1099*x^17 - 7371*x^16 + 51877*x^15 - 28459*x^14 - 763602*x^13 + 2834454*x^12 - 3354876*x^11 - 12561948*x^10 - 9581544*x^9 + 867775608*x^8 - 2671533936*x^7 + 2861475120*x^6 - 10704143520*x^5 + 18597509280*x^4 + 4102539840*x^3 + 39186141120*x^2 - 60906867840*x + 19907441280)
 
gp: K = bnfinit(x^21 - 7*x^20 - 21*x^19 + 427*x^18 - 1099*x^17 - 7371*x^16 + 51877*x^15 - 28459*x^14 - 763602*x^13 + 2834454*x^12 - 3354876*x^11 - 12561948*x^10 - 9581544*x^9 + 867775608*x^8 - 2671533936*x^7 + 2861475120*x^6 - 10704143520*x^5 + 18597509280*x^4 + 4102539840*x^3 + 39186141120*x^2 - 60906867840*x + 19907441280, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} - 21 x^{19} + 427 x^{18} - 1099 x^{17} - 7371 x^{16} + 51877 x^{15} - 28459 x^{14} - 763602 x^{13} + 2834454 x^{12} - 3354876 x^{11} - 12561948 x^{10} - 9581544 x^{9} + 867775608 x^{8} - 2671533936 x^{7} + 2861475120 x^{6} - 10704143520 x^{5} + 18597509280 x^{4} + 4102539840 x^{3} + 39186141120 x^{2} - 60906867840 x + 19907441280 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[1, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(385299031450283695081389016328751512292325294080000000000000000000=2^{33}\cdot 3^{19}\cdot 5^{19}\cdot 7^{21}\cdot 167^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1327.80$
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:  $2, 3, 5, 7, 167$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{8} + \frac{1}{6} a^{7} + \frac{1}{6} a^{5} + \frac{1}{12} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{9} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{4} a^{5} - \frac{1}{6} a^{4}$, $\frac{1}{12} a^{10} - \frac{1}{6} a^{7} - \frac{1}{4} a^{6} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{11} + \frac{1}{12} a^{7} + \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{24} a^{12} - \frac{1}{24} a^{10} - \frac{1}{24} a^{8} + \frac{1}{6} a^{7} + \frac{5}{24} a^{6} + \frac{1}{6} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{24} a^{13} - \frac{1}{24} a^{11} - \frac{1}{24} a^{9} - \frac{1}{8} a^{7} + \frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{6} a^{4}$, $\frac{1}{360} a^{14} - \frac{1}{360} a^{13} - \frac{1}{60} a^{12} - \frac{1}{72} a^{11} - \frac{1}{36} a^{10} - \frac{1}{120} a^{9} - \frac{1}{180} a^{8} - \frac{7}{360} a^{7} + \frac{1}{8} a^{6} + \frac{1}{6} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{720} a^{15} - \frac{1}{720} a^{14} + \frac{1}{80} a^{13} - \frac{1}{144} a^{12} - \frac{5}{144} a^{11} - \frac{1}{240} a^{10} - \frac{17}{720} a^{9} - \frac{7}{720} a^{8} - \frac{1}{4} a^{7} + \frac{1}{6} a^{6} + \frac{1}{12} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{1440} a^{16} - \frac{1}{1440} a^{15} - \frac{1}{1440} a^{14} - \frac{5}{288} a^{13} + \frac{1}{288} a^{12} + \frac{17}{1440} a^{11} + \frac{53}{1440} a^{10} + \frac{53}{1440} a^{9} - \frac{1}{144} a^{8} - \frac{13}{72} a^{7} + \frac{1}{6} a^{6} + \frac{1}{12} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4320} a^{17} - \frac{1}{4320} a^{16} - \frac{1}{1440} a^{15} + \frac{1}{4320} a^{14} + \frac{83}{4320} a^{13} + \frac{7}{480} a^{12} - \frac{137}{4320} a^{11} + \frac{119}{4320} a^{10} - \frac{7}{180} a^{9} + \frac{7}{240} a^{8} + \frac{43}{180} a^{7} + \frac{1}{24} a^{6} + \frac{1}{12} a^{5} - \frac{1}{4} a^{4} - \frac{1}{6} a^{3}$, $\frac{1}{8640} a^{18} - \frac{1}{8640} a^{17} - \frac{1}{2880} a^{16} + \frac{1}{8640} a^{15} - \frac{1}{8640} a^{14} + \frac{49}{2880} a^{13} - \frac{173}{8640} a^{12} + \frac{179}{8640} a^{11} + \frac{11}{720} a^{10} + \frac{1}{480} a^{9} + \frac{5}{144} a^{8} + \frac{47}{360} a^{7} + \frac{1}{12} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{43200} a^{19} - \frac{1}{43200} a^{18} - \frac{1}{43200} a^{17} + \frac{1}{8640} a^{16} - \frac{1}{1728} a^{15} + \frac{59}{43200} a^{14} - \frac{169}{43200} a^{13} - \frac{109}{43200} a^{12} + \frac{73}{2160} a^{11} - \frac{167}{4320} a^{10} - \frac{77}{2400} a^{9} + \frac{43}{3600} a^{8} + \frac{199}{1800} a^{7} - \frac{7}{60} a^{6} + \frac{1}{60} a^{5} + \frac{1}{12} a^{4} + \frac{1}{6} a^{3} - \frac{1}{10} a^{2} + \frac{1}{10} a - \frac{2}{5}$, $\frac{1}{1897591913341886541473143274118096733401882890407322765319039400040594905411200} a^{20} - \frac{11024950176647895600783290524472886341779059565164292538542547430792691447}{1897591913341886541473143274118096733401882890407322765319039400040594905411200} a^{19} - \frac{1245285364278433719284992897177412111939907545648407417238579516670529459}{29193721743721331407279127294124565129259736775497273312600606154470690852480} a^{18} + \frac{109119237974003524777401881926612712010287807034797307887810464205380456621}{1897591913341886541473143274118096733401882890407322765319039400040594905411200} a^{17} - \frac{4091916703376184998120942904052554755003287790859602293096123667325919109}{75903676533675461658925730964723869336075315616292910612761576001623796216448} a^{16} + \frac{303928761977116170535186232695675704217603247333428061870346886163293911499}{1897591913341886541473143274118096733401882890407322765319039400040594905411200} a^{15} + \frac{869984238638739572331714736914658788275129112508129650385135045271206574767}{1897591913341886541473143274118096733401882890407322765319039400040594905411200} a^{14} - \frac{468849322118253702466505339177023733473256097377949525940298263635129365081}{75903676533675461658925730964723869336075315616292910612761576001623796216448} a^{13} - \frac{1597431761679759438285799677899988206598938854952106402981714743214665376787}{237198989167735817684142909264762091675235361300915345664879925005074363176400} a^{12} - \frac{23291652688923812122438378192422290875083720276846077108901348699523671897}{878514774695317843274603367647267006204575412225612391351407129648423567320} a^{11} - \frac{2343934356411636197314984681426446927787022854678725603185784526262304464723}{59299747291933954421035727316190522918808840325228836416219981251268590794100} a^{10} - \frac{121100978565159844896987646666734678536149117512947529799395754682589026909}{9883291215322325736839287886031753819801473387538139402703330208544765132350} a^{9} - \frac{213439756172974966412008958227904777585752615360920171783002064679393415431}{15813265944515721178942860617650806111682357420061023044325328333671624211760} a^{8} + \frac{4268235733391828504681461130704715967989900162547877361100759349207612086739}{19766582430644651473678575772063507639602946775076278805406660417089530264700} a^{7} + \frac{71317623728511210224756050772432220536744487963235110037198082838363780916}{329443040510744191227976262867725127326715779584604646756777673618158837745} a^{6} + \frac{289406433611418139591625186044008512332099984987313967544329712659524174209}{2635544324085953529823810102941801018613726236676837174054221388945270701960} a^{5} + \frac{19325367881875422581839196867728723715851891339587086750388591979319154271}{131777216204297676491190505147090050930686311833841858702711069447263535098} a^{4} + \frac{49205223894097134006996033124608967667579120746108306503620202207778317703}{329443040510744191227976262867725127326715779584604646756777673618158837745} a^{3} + \frac{48554374013288449924794380434581906326915123673780532328482432519309738418}{109814346836914730409325420955908375775571926528201548918925891206052945915} a^{2} - \frac{6149175508624318795701046862584269834794591232098450419623702003514144842}{21962869367382946081865084191181675155114385305640309783785178241210589183} a + \frac{31006737318900635707356992809316703272188463726602519398542991358720953771}{109814346836914730409325420955908375775571926528201548918925891206052945915}$

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

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $10$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times F_7$ (as 21T15):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 252
The 21 conjugacy class representatives for $S_3\times F_7$
Character table for $S_3\times F_7$ is not computed

Intermediate fields

3.1.20040.1, 7.1.600362847000000.22

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

Sibling fields

Degree 42 siblings: 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 R R R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
$2$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.14.27.121$x^{14} - 4 x^{13} - 4 x^{12} + 8 x^{11} + 8 x^{10} - 6 x^{8} + 8 x^{7} + 6 x^{6} - 4 x^{5} - 2 x^{4} - 4 x^{3} + 4 x^{2} + 8 x + 6$$14$$1$$27$$(C_7:C_3) \times C_2$$[3]_{7}^{3}$
$3$3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
3.14.13.1$x^{14} - 3$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$
$5$5.7.6.1$x^{7} - 5$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
5.14.13.2$x^{14} + 10$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$
7Data not computed
$167$$\Q_{167}$$x + 2$$1$$1$$0$Trivial$[\ ]$
167.2.1.1$x^{2} - 167$$2$$1$$1$$C_2$$[\ ]_{2}$
167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
167.4.2.1$x^{4} + 1503 x^{2} + 697225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
167.4.2.1$x^{4} + 1503 x^{2} + 697225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
167.4.2.1$x^{4} + 1503 x^{2} + 697225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$