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Properties

Label 21.3.72011138722...2983.1
Degree $21$
Signature $[3, 9]$
Discriminant $-\,3^{18}\cdot 7^{17}\cdot 19^{14}$
Root discriminant $88.22$
Ramified primes $3, 7, 19$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3\times F_7$ (as 21T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-10983749, 29642690, 11077409, 23180423, -34239268, -8676262, 5924631, 12752578, -9563747, 736266, 1254566, -327079, -52788, 27157, 1633, 513, -1375, 8, 149, -10, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - 10*x^19 + 149*x^18 + 8*x^17 - 1375*x^16 + 513*x^15 + 1633*x^14 + 27157*x^13 - 52788*x^12 - 327079*x^11 + 1254566*x^10 + 736266*x^9 - 9563747*x^8 + 12752578*x^7 + 5924631*x^6 - 8676262*x^5 - 34239268*x^4 + 23180423*x^3 + 11077409*x^2 + 29642690*x - 10983749)
 
gp: K = bnfinit(x^21 - 7*x^20 - 10*x^19 + 149*x^18 + 8*x^17 - 1375*x^16 + 513*x^15 + 1633*x^14 + 27157*x^13 - 52788*x^12 - 327079*x^11 + 1254566*x^10 + 736266*x^9 - 9563747*x^8 + 12752578*x^7 + 5924631*x^6 - 8676262*x^5 - 34239268*x^4 + 23180423*x^3 + 11077409*x^2 + 29642690*x - 10983749, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} - 10 x^{19} + 149 x^{18} + 8 x^{17} - 1375 x^{16} + 513 x^{15} + 1633 x^{14} + 27157 x^{13} - 52788 x^{12} - 327079 x^{11} + 1254566 x^{10} + 736266 x^{9} - 9563747 x^{8} + 12752578 x^{7} + 5924631 x^{6} - 8676262 x^{5} - 34239268 x^{4} + 23180423 x^{3} + 11077409 x^{2} + 29642690 x - 10983749 \)

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:  \(-72011138722425633565208857577376396922983=-\,3^{18}\cdot 7^{17}\cdot 19^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $88.22$
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:  $3, 7, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{2}{9} a^{8} + \frac{1}{3} a^{7} - \frac{1}{9} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{9} a^{2} + \frac{2}{9}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{13} + \frac{1}{3} a^{10} + \frac{1}{9} a^{9} - \frac{1}{9} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{2}{9} a^{3} + \frac{1}{3} a^{2} + \frac{2}{9} a - \frac{1}{3}$, $\frac{1}{9} a^{16} - \frac{1}{9} a^{12} + \frac{4}{9} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{9}$, $\frac{1}{9} a^{17} - \frac{1}{9} a^{13} + \frac{1}{9} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{9} a^{7} + \frac{1}{3} a^{6} + \frac{4}{9} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{189} a^{18} - \frac{1}{189} a^{17} - \frac{1}{27} a^{16} - \frac{2}{63} a^{15} - \frac{8}{189} a^{14} + \frac{1}{27} a^{13} - \frac{2}{63} a^{12} - \frac{1}{27} a^{11} - \frac{1}{189} a^{10} - \frac{10}{21} a^{9} - \frac{11}{27} a^{8} - \frac{50}{189} a^{7} + \frac{19}{63} a^{6} + \frac{11}{27} a^{5} - \frac{10}{27} a^{4} - \frac{1}{9} a^{3} + \frac{10}{27} a^{2} - \frac{2}{27} a - \frac{10}{27}$, $\frac{1}{189} a^{19} - \frac{8}{189} a^{17} + \frac{8}{189} a^{16} + \frac{1}{27} a^{15} - \frac{1}{189} a^{14} - \frac{20}{189} a^{13} + \frac{29}{189} a^{12} - \frac{8}{189} a^{11} - \frac{1}{27} a^{10} + \frac{43}{189} a^{9} - \frac{64}{189} a^{8} + \frac{7}{27} a^{7} - \frac{76}{189} a^{6} - \frac{8}{27} a^{5} - \frac{10}{27} a^{4} + \frac{10}{27} a^{3} + \frac{8}{27} a^{2} - \frac{2}{9} a + \frac{5}{27}$, $\frac{1}{285115271312775302688431031516495668439755630444097277223054106041736544971} a^{20} + \frac{701781261275273628517945014625489938364146027124109415972498192600766052}{285115271312775302688431031516495668439755630444097277223054106041736544971} a^{19} - \frac{428199515995979347607976927423398463545918063410856059613180724629561341}{285115271312775302688431031516495668439755630444097277223054106041736544971} a^{18} - \frac{71717971569490786338604246107077735044263641679207757664255513120048763}{1508546409062303188827677415431194012908759949439668133455312730379558439} a^{17} + \frac{3002517411515084566767314605465861025874272612526074636004042120818300230}{285115271312775302688431031516495668439755630444097277223054106041736544971} a^{16} - \frac{9444595121815558569214080942989164705137439686583883292918275073408813353}{285115271312775302688431031516495668439755630444097277223054106041736544971} a^{15} - \frac{62059215948829694792507975633424097919186581940776281075621111135977666}{1508546409062303188827677415431194012908759949439668133455312730379558439} a^{14} + \frac{27094830126966476698845615459462977497472225101131419107286757814051008347}{285115271312775302688431031516495668439755630444097277223054106041736544971} a^{13} + \frac{6475734660740292207984580050897968086738533532058435424960361661953669016}{40730753044682186098347290216642238348536518634871039603293443720248077853} a^{12} + \frac{5563588686346260810686642897367419497290310898905205465004857596273918535}{95038423770925100896143677172165222813251876814699092407684702013912181657} a^{11} + \frac{22213742003869202933606464243069682149901271798978464165855145252809851354}{285115271312775302688431031516495668439755630444097277223054106041736544971} a^{10} + \frac{4249986636871266739548266956676951666604398552518044705036704743308443194}{40730753044682186098347290216642238348536518634871039603293443720248077853} a^{9} - \frac{5596861223974940955485520960947345690215107657291835701486510720329818508}{31679474590308366965381225724055074271083958938233030802561567337970727219} a^{8} - \frac{2377352080108251349618289415045885204159318726657956429521098193193847781}{285115271312775302688431031516495668439755630444097277223054106041736544971} a^{7} - \frac{38026324681538180709735410063594887289319395944136039325547064469722463703}{285115271312775302688431031516495668439755630444097277223054106041736544971} a^{6} - \frac{3490547994343042097847954609420605872595603733955350974346396433667171171}{13576917681560728699449096738880746116178839544957013201097814573416025951} a^{5} + \frac{18297830045243406843866096279200027719153762846121943084348546505511016110}{40730753044682186098347290216642238348536518634871039603293443720248077853} a^{4} + \frac{318851002588580172678822102599164340714353099454174628491817234323259988}{40730753044682186098347290216642238348536518634871039603293443720248077853} a^{3} - \frac{7425028392051553645389813138236170774324052140032077676060099756904413796}{40730753044682186098347290216642238348536518634871039603293443720248077853} a^{2} - \frac{4727405112452063353200562813241399335276985404350494646049257871761830936}{13576917681560728699449096738880746116178839544957013201097814573416025951} a + \frac{1821325369268335683306127736940006049958797367677652141877173732357898266}{4525639227186909566483032246293582038726279848319004400365938191138675317}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 1924259099008.9683 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times F_7$ (as 21T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 126
The 21 conjugacy class representatives for $C_3\times F_7$
Character table for $C_3\times F_7$ is not computed

Intermediate fields

3.3.17689.1, 7.1.1596732379263.1

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

Sibling fields

Degree 21 siblings: 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 $21$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
$3$3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
$7$7.3.2.2$x^{3} - 7$$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}$
19Data not computed