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Properties

Label 21.1.56975258045...7744.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{18}\cdot 7^{36}\cdot 31^{10}$
Root discriminant $261.18$
Ramified primes $2, 7, 31$
Class number $7$ (GRH)
Class group $[7]$ (GRH)
Galois group $C_7^2:S_3$ (as 21T17)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-50542023253032, -17560545695056, -3218488878808, -4154662343608, -2346556141272, -418374125568, 11054922536, -1691077124, -4000275328, 22521464, 291794048, 74605944, 16100140, -59192, -1495990, -232820, 38318, 9422, -350, -154, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 154*x^19 - 350*x^18 + 9422*x^17 + 38318*x^16 - 232820*x^15 - 1495990*x^14 - 59192*x^13 + 16100140*x^12 + 74605944*x^11 + 291794048*x^10 + 22521464*x^9 - 4000275328*x^8 - 1691077124*x^7 + 11054922536*x^6 - 418374125568*x^5 - 2346556141272*x^4 - 4154662343608*x^3 - 3218488878808*x^2 - 17560545695056*x - 50542023253032)
 
gp: K = bnfinit(x^21 - 154*x^19 - 350*x^18 + 9422*x^17 + 38318*x^16 - 232820*x^15 - 1495990*x^14 - 59192*x^13 + 16100140*x^12 + 74605944*x^11 + 291794048*x^10 + 22521464*x^9 - 4000275328*x^8 - 1691077124*x^7 + 11054922536*x^6 - 418374125568*x^5 - 2346556141272*x^4 - 4154662343608*x^3 - 3218488878808*x^2 - 17560545695056*x - 50542023253032, 1)
 

Normalized defining polynomial

\( x^{21} - 154 x^{19} - 350 x^{18} + 9422 x^{17} + 38318 x^{16} - 232820 x^{15} - 1495990 x^{14} - 59192 x^{13} + 16100140 x^{12} + 74605944 x^{11} + 291794048 x^{10} + 22521464 x^{9} - 4000275328 x^{8} - 1691077124 x^{7} + 11054922536 x^{6} - 418374125568 x^{5} - 2346556141272 x^{4} - 4154662343608 x^{3} - 3218488878808 x^{2} - 17560545695056 x - 50542023253032 \)

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:  \(569752580450182296234409142462680787211686353567744=2^{18}\cdot 7^{36}\cdot 31^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $261.18$
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:  $2, 7, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{14}$, $\frac{1}{4} a^{15}$, $\frac{1}{4} a^{16}$, $\frac{1}{8} a^{17} - \frac{1}{2} a^{3}$, $\frac{1}{24} a^{18} - \frac{1}{12} a^{16} + \frac{1}{12} a^{15} - \frac{1}{12} a^{14} - \frac{1}{12} a^{13} + \frac{1}{12} a^{12} - \frac{1}{6} a^{11} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{408} a^{19} + \frac{1}{68} a^{18} + \frac{2}{51} a^{17} - \frac{7}{102} a^{16} - \frac{13}{204} a^{15} - \frac{4}{51} a^{14} + \frac{1}{51} a^{13} + \frac{5}{102} a^{12} + \frac{3}{17} a^{11} - \frac{4}{51} a^{10} + \frac{5}{102} a^{9} - \frac{23}{102} a^{8} + \frac{11}{51} a^{7} + \frac{6}{17} a^{6} + \frac{49}{102} a^{5} - \frac{2}{17} a^{4} + \frac{16}{51} a^{3} + \frac{5}{51} a^{2} + \frac{7}{17} a - \frac{1}{17}$, $\frac{1}{1455426143404991616695356644999181054150559900428496719549565151654844941344056953045369450999448521896833951308124038696} a^{20} + \frac{358099488039695327424811358861358423146508383125810617184718179134306369227504774826659159745975935120752505839288325}{1455426143404991616695356644999181054150559900428496719549565151654844941344056953045369450999448521896833951308124038696} a^{19} + \frac{2337102345466173329767369938586161330748157595156740752729895277918797025634806088737885531911855055974742644579438245}{485142047801663872231785548333060351383519966809498906516521717218281647114685651015123150333149507298944650436041346232} a^{18} + \frac{77921144163891295840079362806573493186095020280274381229733981654890358884255931606570292223424812495217222382726765243}{1455426143404991616695356644999181054150559900428496719549565151654844941344056953045369450999448521896833951308124038696} a^{17} + \frac{4845044390311646111886856634150825035235862830462672209888131578327683175375120581190722294326271409587551097747080953}{727713071702495808347678322499590527075279950214248359774782575827422470672028476522684725499724260948416975654062019348} a^{16} - \frac{20137700860786084213205006895166098711481604361429918686853258013099215700479293566225244737946301697409780006714784895}{727713071702495808347678322499590527075279950214248359774782575827422470672028476522684725499724260948416975654062019348} a^{15} - \frac{65948056096770592904919314334831416463420688713335068186645901185231988590271636318276862793404452202241212483346895}{26952335988981326235099197129614463965751109267194383695362317623237869284149202834173508351841639294385813913113408124} a^{14} + \frac{21908876564307580307564824606836628681204047870644177779781462975910790464973656319572841343806749408997669926751560397}{727713071702495808347678322499590527075279950214248359774782575827422470672028476522684725499724260948416975654062019348} a^{13} - \frac{87223518686748732374586543086867709881617207061233792473630292377507584177103576914422976934664063440255727354393212539}{727713071702495808347678322499590527075279950214248359774782575827422470672028476522684725499724260948416975654062019348} a^{12} + \frac{30899403758832394134620823495434797464857932405702615964152555796018551172966677312295767153931300410864503761064371197}{181928267925623952086919580624897631768819987553562089943695643956855617668007119130671181374931065237104243913515504837} a^{11} - \frac{51163471885043103523911335036450513237992666402907944317541610029289482790914085716304536939612495863410492980329237797}{363856535851247904173839161249795263537639975107124179887391287913711235336014238261342362749862130474208487827031009674} a^{10} - \frac{2760446783334215437048557173651978910569459785918468701090605953861280961843797566428855615301468850804787846972664469}{363856535851247904173839161249795263537639975107124179887391287913711235336014238261342362749862130474208487827031009674} a^{9} - \frac{42275464283051929938124533674622041502179927927211697561235480509526378416748755198137102378663754391535134226285323782}{181928267925623952086919580624897631768819987553562089943695643956855617668007119130671181374931065237104243913515504837} a^{8} + \frac{53678232569163724863689346876200968411558781252345542371368380187121104518778199304183048359978111046984701619358004073}{363856535851247904173839161249795263537639975107124179887391287913711235336014238261342362749862130474208487827031009674} a^{7} - \frac{2703530143524124981223290937875895684829756574634751006910758243325087731032350606057149563831954844180726659472412028}{20214251991735994676324397847210847974313331950395787771521738217428401963111902125630131263881229470789360434835056093} a^{6} + \frac{54436776300816158786337285887624861214341219691954633206552874652842187279385757136166984604919909682876716108837864411}{181928267925623952086919580624897631768819987553562089943695643956855617668007119130671181374931065237104243913515504837} a^{5} + \frac{51453730735959927031207716050455147065337976867729986541172774938000863452612854183541503896885878079258905327490147603}{363856535851247904173839161249795263537639975107124179887391287913711235336014238261342362749862130474208487827031009674} a^{4} - \frac{71692009969161146314528664497360909127295639304144930340892950927757806138195960935211616796287004361800597214471373061}{363856535851247904173839161249795263537639975107124179887391287913711235336014238261342362749862130474208487827031009674} a^{3} + \frac{7788389321998256353046134103652299174394161156299011224349057719901242612503357068779092272530322384155976667304449817}{60642755975207984028973193541632543922939995851187363314565214652285205889335706376890393791643688412368081304505168279} a^{2} - \frac{7118034879590876688382994612650961393667852306343181494938682302476850422632321200457607152894298499786310080336092392}{16538933447783995644265416420445239251710907959414735449426876723350510697091556284606471034084642294282203992137773167} a - \frac{23364281511151157492032719565054763955991218648285438397470631103322279270700993460516569626494367810223103395530193}{1189073646572705569195552814541814586724313644140928692442455189260494233124229536801772427287131145340550613813826829}$

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:  $10$
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:  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:  \( 21312988696200000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_7^2:S_3$ (as 21T17):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 294
The 20 conjugacy class representatives for $C_7^2:S_3$
Character table for $C_7^2:S_3$

Intermediate fields

3.1.31.1

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

Sibling fields

Degree 14 sibling: data not computed
Degree 21 sibling: data not computed
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 ${\href{/LocalNumberField/3.14.0.1}{14} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/11.14.0.1}{14} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.14.0.1}{14} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.14.0.1}{14} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/23.14.0.1}{14} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.14.0.1}{14} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ R ${\href{/LocalNumberField/37.14.0.1}{14} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/53.14.0.1}{14} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{7}$

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
2Data not computed
7Data not computed
$31$$\Q_{31}$$x + 7$$1$$1$$0$Trivial$[\ ]$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.14.7.2$x^{14} - 887503681 x^{2} + 495227053998$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$