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Properties

Label 21.5.38225159209...2816.1
Degree $21$
Signature $[5, 8]$
Discriminant $2^{36}\cdot 11^{19}\cdot 71^{7}$
Root discriminant $118.95$
Ramified primes $2, 11, 71$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T85

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-162458072, 333108324, -413258960, 265754960, -141705136, 54407188, 4239088, -15644640, 11243466, -6659517, 2325392, -451160, 52730, 54335, -32524, 5320, -902, -71, 220, -32, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^20 - 32*x^19 + 220*x^18 - 71*x^17 - 902*x^16 + 5320*x^15 - 32524*x^14 + 54335*x^13 + 52730*x^12 - 451160*x^11 + 2325392*x^10 - 6659517*x^9 + 11243466*x^8 - 15644640*x^7 + 4239088*x^6 + 54407188*x^5 - 141705136*x^4 + 265754960*x^3 - 413258960*x^2 + 333108324*x - 162458072)
 
gp: K = bnfinit(x^21 - 6*x^20 - 32*x^19 + 220*x^18 - 71*x^17 - 902*x^16 + 5320*x^15 - 32524*x^14 + 54335*x^13 + 52730*x^12 - 451160*x^11 + 2325392*x^10 - 6659517*x^9 + 11243466*x^8 - 15644640*x^7 + 4239088*x^6 + 54407188*x^5 - 141705136*x^4 + 265754960*x^3 - 413258960*x^2 + 333108324*x - 162458072, 1)
 

Normalized defining polynomial

\( x^{21} - 6 x^{20} - 32 x^{19} + 220 x^{18} - 71 x^{17} - 902 x^{16} + 5320 x^{15} - 32524 x^{14} + 54335 x^{13} + 52730 x^{12} - 451160 x^{11} + 2325392 x^{10} - 6659517 x^{9} + 11243466 x^{8} - 15644640 x^{7} + 4239088 x^{6} + 54407188 x^{5} - 141705136 x^{4} + 265754960 x^{3} - 413258960 x^{2} + 333108324 x - 162458072 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(38225159209415003799521096455077448244002816=2^{36}\cdot 11^{19}\cdot 71^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $118.95$
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:  $2, 11, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{19} - \frac{1}{4} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7}$, $\frac{1}{912513653473656343486771612236835316881016661930564432537440127863044616} a^{20} - \frac{49537078659084543533730326498691300139451762407201688718594504798682521}{456256826736828171743385806118417658440508330965282216268720063931522308} a^{19} - \frac{3764102016710297291601255198624197191376591867652028419177247271027531}{114064206684207042935846451529604414610127082741320554067180015982880577} a^{18} - \frac{11609676498197244108345302413179135587756036394932477871501030730868024}{114064206684207042935846451529604414610127082741320554067180015982880577} a^{17} - \frac{84286361725860263659306269359515168480717570050472508811821382152899273}{912513653473656343486771612236835316881016661930564432537440127863044616} a^{16} - \frac{48860880286196999830462911156304171586150771652914092046162080977121489}{456256826736828171743385806118417658440508330965282216268720063931522308} a^{15} - \frac{7916926546922285700802275857358350588890025291855943542219708572149845}{114064206684207042935846451529604414610127082741320554067180015982880577} a^{14} + \frac{22824922224302722920773129778867150829039004457804056158088862604340455}{456256826736828171743385806118417658440508330965282216268720063931522308} a^{13} - \frac{199935680930845531773181832665989783140954579262566806437903329727858583}{912513653473656343486771612236835316881016661930564432537440127863044616} a^{12} - \frac{4857083052821038111293105175858487430443885653789516181298583758697425}{456256826736828171743385806118417658440508330965282216268720063931522308} a^{11} + \frac{24818835600546850228682634310473618530122798891944054525482918409588741}{114064206684207042935846451529604414610127082741320554067180015982880577} a^{10} + \frac{12932979600023984232822931300449774772024639010043489347612617196779179}{228128413368414085871692903059208829220254165482641108134360031965761154} a^{9} + \frac{132916211058994940742001267296746382067495894592513978224653301878688649}{912513653473656343486771612236835316881016661930564432537440127863044616} a^{8} - \frac{87241985736551296917297824017502981098991613956315604480712394197254893}{456256826736828171743385806118417658440508330965282216268720063931522308} a^{7} + \frac{14177726963944418997756472388436460389938054579401018872679072934042507}{114064206684207042935846451529604414610127082741320554067180015982880577} a^{6} - \frac{176320234889427991557269549032854964968734233995237618597395666184840473}{456256826736828171743385806118417658440508330965282216268720063931522308} a^{5} - \frac{32901826255744937776860103817614669600684646361530714557535764903261231}{456256826736828171743385806118417658440508330965282216268720063931522308} a^{4} - \frac{10732967242555960996432513457336948051728770113260580015720520407319323}{228128413368414085871692903059208829220254165482641108134360031965761154} a^{3} - \frac{50113262680930244783032263141870119815209106595934313079731467380437529}{114064206684207042935846451529604414610127082741320554067180015982880577} a^{2} + \frac{66269187551103553512156910141964748161209065281880822603851528924450055}{228128413368414085871692903059208829220254165482641108134360031965761154} a + \frac{51922734391326434585754477034112572755478102267551795833715983287099726}{114064206684207042935846451529604414610127082741320554067180015982880577}$

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:  $12$
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:  \( 2512137314520000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T85:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 40320
The 14 conjugacy class representatives for t21n85
Character table for t21n85

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 42 siblings: data not computed
Arithmetically equvalently 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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.14.0.1}{14} }{,}\,{\href{/LocalNumberField/19.7.0.1}{7} }$ ${\href{/LocalNumberField/23.14.0.1}{14} }{,}\,{\href{/LocalNumberField/23.7.0.1}{7} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.14.0.1}{14} }{,}\,{\href{/LocalNumberField/59.7.0.1}{7} }$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.4.8.8$x^{4} + 4 x + 2$$4$$1$$8$$S_4$$[8/3, 8/3]_{3}^{2}$
2.4.8.8$x^{4} + 4 x + 2$$4$$1$$8$$S_4$$[8/3, 8/3]_{3}^{2}$
2.6.10.2$x^{6} + 2 x^{5} + 2 x^{4} + 2 x^{2} + 2$$6$$1$$10$$S_4$$[8/3, 8/3]_{3}^{2}$
2.6.10.3$x^{6} + 2 x^{5} + 4 x + 2$$6$$1$$10$$S_4$$[8/3, 8/3]_{3}^{2}$
11Data not computed
$71$71.7.0.1$x^{7} - x + 6$$1$$7$$0$$C_7$$[\ ]^{7}$
71.14.7.2$x^{14} - 128100283921 x^{2} + 54570720950346$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$