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Properties

Label 21.1.23799537214...9392.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{33}\cdot 3^{18}\cdot 7^{40}\cdot 379^{7}$
Root discriminant $2245.25$
Ramified primes $2, 3, 7, 379$
Class number Not computed
Class group Not computed
Galois group $S_3\times F_7$ (as 21T15)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-35029115785232, -61556969399632, -50296288035280, -22166544339184, -3757986589840, 1150521936032, 784511093184, 150044092120, -15880653152, -13338735312, -1890715204, 289386524, 123851322, 7171822, -2971235, -503307, 22673, 11501, 287, -133, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - 133*x^19 + 287*x^18 + 11501*x^17 + 22673*x^16 - 503307*x^15 - 2971235*x^14 + 7171822*x^13 + 123851322*x^12 + 289386524*x^11 - 1890715204*x^10 - 13338735312*x^9 - 15880653152*x^8 + 150044092120*x^7 + 784511093184*x^6 + 1150521936032*x^5 - 3757986589840*x^4 - 22166544339184*x^3 - 50296288035280*x^2 - 61556969399632*x - 35029115785232)
 
gp: K = bnfinit(x^21 - 7*x^20 - 133*x^19 + 287*x^18 + 11501*x^17 + 22673*x^16 - 503307*x^15 - 2971235*x^14 + 7171822*x^13 + 123851322*x^12 + 289386524*x^11 - 1890715204*x^10 - 13338735312*x^9 - 15880653152*x^8 + 150044092120*x^7 + 784511093184*x^6 + 1150521936032*x^5 - 3757986589840*x^4 - 22166544339184*x^3 - 50296288035280*x^2 - 61556969399632*x - 35029115785232, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} - 133 x^{19} + 287 x^{18} + 11501 x^{17} + 22673 x^{16} - 503307 x^{15} - 2971235 x^{14} + 7171822 x^{13} + 123851322 x^{12} + 289386524 x^{11} - 1890715204 x^{10} - 13338735312 x^{9} - 15880653152 x^{8} + 150044092120 x^{7} + 784511093184 x^{6} + 1150521936032 x^{5} - 3757986589840 x^{4} - 22166544339184 x^{3} - 50296288035280 x^{2} - 61556969399632 x - 35029115785232 \)

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:  \(23799537214144544855987852949932308494594769678441899396662782058299392=2^{33}\cdot 3^{18}\cdot 7^{40}\cdot 379^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2245.25$
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:  $2, 3, 7, 379$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{588} a^{14} + \frac{5}{28} a^{13} + \frac{11}{84} a^{12} - \frac{1}{28} a^{11} + \frac{3}{28} a^{10} + \frac{1}{28} a^{9} - \frac{5}{84} a^{8} - \frac{9}{196} a^{7} - \frac{10}{21} a^{6} - \frac{3}{14} a^{5} + \frac{5}{14} a^{4} - \frac{2}{7} a^{3} + \frac{1}{21} a^{2} + \frac{3}{7} a - \frac{40}{147}$, $\frac{1}{1764} a^{15} - \frac{1}{1764} a^{14} + \frac{59}{252} a^{13} + \frac{7}{36} a^{12} - \frac{17}{84} a^{11} + \frac{5}{84} a^{10} + \frac{13}{252} a^{9} - \frac{139}{1764} a^{8} - \frac{16}{441} a^{7} - \frac{5}{63} a^{6} - \frac{10}{21} a^{5} - \frac{1}{21} a^{4} - \frac{2}{9} a^{3} + \frac{8}{63} a^{2} + \frac{191}{441} a - \frac{170}{441}$, $\frac{1}{1764} a^{16} + \frac{1}{1764} a^{14} - \frac{1}{28} a^{13} + \frac{13}{252} a^{12} - \frac{1}{4} a^{11} - \frac{17}{252} a^{10} + \frac{47}{588} a^{9} + \frac{5}{126} a^{8} + \frac{103}{588} a^{7} + \frac{23}{126} a^{6} - \frac{1}{6} a^{5} - \frac{25}{126} a^{4} + \frac{1}{21} a^{3} + \frac{16}{441} a^{2} + \frac{1}{3} a - \frac{47}{441}$, $\frac{1}{3528} a^{17} - \frac{1}{3528} a^{16} - \frac{1}{3528} a^{15} + \frac{1}{3528} a^{14} + \frac{31}{168} a^{13} - \frac{37}{168} a^{12} + \frac{85}{504} a^{11} + \frac{491}{3528} a^{10} + \frac{47}{882} a^{9} - \frac{403}{1764} a^{8} + \frac{181}{588} a^{7} + \frac{17}{42} a^{6} - \frac{5}{126} a^{5} + \frac{53}{126} a^{4} + \frac{191}{882} a^{3} - \frac{211}{441} a^{2} + \frac{17}{49} a + \frac{4}{49}$, $\frac{1}{3528} a^{18} - \frac{1}{28} a^{13} + \frac{19}{84} a^{12} - \frac{29}{588} a^{11} - \frac{3}{56} a^{10} + \frac{1}{84} a^{9} + \frac{3}{28} a^{8} - \frac{3}{28} a^{7} - \frac{1}{21} a^{6} + \frac{1}{14} a^{5} - \frac{24}{49} a^{4} - \frac{1}{14} a^{3} - \frac{2}{7} a^{2} + \frac{1}{21} a - \frac{23}{63}$, $\frac{1}{24696} a^{19} - \frac{1}{8232} a^{18} - \frac{1}{12348} a^{17} + \frac{1}{6174} a^{16} - \frac{1}{6174} a^{15} - \frac{1}{1372} a^{14} - \frac{31}{294} a^{13} + \frac{400}{3087} a^{12} + \frac{25}{24696} a^{11} - \frac{5399}{24696} a^{10} - \frac{2651}{12348} a^{9} + \frac{233}{2058} a^{8} - \frac{787}{4116} a^{7} - \frac{80}{441} a^{6} - \frac{5}{6174} a^{5} - \frac{1259}{6174} a^{4} + \frac{311}{6174} a^{3} + \frac{163}{343} a^{2} + \frac{52}{3087} a - \frac{284}{3087}$, $\frac{1}{66681183793121508273901358054936890022263838974800488952176723674932007124086063545367373212788552} a^{20} - \frac{1016339344884395278435586589884374277477748252035221455503642931684098231923303660256871038779}{66681183793121508273901358054936890022263838974800488952176723674932007124086063545367373212788552} a^{19} + \frac{305930351088608922924913505561130032861503842140882646634798584203037654939364386073007319027}{7409020421457945363766817561659654446918204330533387661352969297214667458231784838374152579198728} a^{18} - \frac{223111577331752212204970168030449367312322442670850044402868082392951408286427403940237222007}{16670295948280377068475339513734222505565959743700122238044180918733001781021515886341843303197138} a^{17} - \frac{175201034908137418844819892289947315270370132607796806859448923946973878946735140275823165279}{2381470849754339581210762787676317500795137106242874605434882988390428825860216555191691900456734} a^{16} - \frac{867071592386720910464710224989331423375960349445825403498442808864100470299241772941054154685}{3704510210728972681883408780829827223459102165266693830676484648607333729115892419187076289599364} a^{15} - \frac{2871961394312348382016690754490728527276202563404048679247679738707010951950228561502742737683}{3704510210728972681883408780829827223459102165266693830676484648607333729115892419187076289599364} a^{14} + \frac{688436269648122130236714704617490285810715623834032032483687281765686208500972585848541681311442}{8335147974140188534237669756867111252782979871850061119022090459366500890510757943170921651598569} a^{13} + \frac{8935403212891875801761723563500054144682787403594906466496142625826691142020448884619445583978289}{66681183793121508273901358054936890022263838974800488952176723674932007124086063545367373212788552} a^{12} - \frac{6984227788401502758526810808406948622916665088403786888382276488191262536780411705829474170207867}{66681183793121508273901358054936890022263838974800488952176723674932007124086063545367373212788552} a^{11} - \frac{11437136113718010751363686482878739456832529687922503553421742503028151058642659234723132456445157}{66681183793121508273901358054936890022263838974800488952176723674932007124086063545367373212788552} a^{10} + \frac{28975354552509658346185230999932442575713088462845943383208558639699061916218756692518153694119}{264607872194926620134529198630701944532793011804763845048320332043380980651135172799076877828526} a^{9} - \frac{1680769101967902799040583654219024596850292822445373586691647940209479694850442111418800157872923}{11113530632186918045650226342489481670377306495800081492029453945822001187347677257561228868798092} a^{8} + \frac{5869864427145179734577504834154684212151719409364180201137218594866773345229832581803182420787689}{16670295948280377068475339513734222505565959743700122238044180918733001781021515886341843303197138} a^{7} - \frac{7611674571522226712181577089979758230620535770773919237708537766620398304857481793446155310397201}{16670295948280377068475339513734222505565959743700122238044180918733001781021515886341843303197138} a^{6} + \frac{7859892543245066651677463766521441050635337272538876864374557154645457967271047785351154019092089}{16670295948280377068475339513734222505565959743700122238044180918733001781021515886341843303197138} a^{5} - \frac{3314941562187923811942877562160256339478313230082068567823810156727681290927674204244704437419811}{16670295948280377068475339513734222505565959743700122238044180918733001781021515886341843303197138} a^{4} - \frac{892797324969357499964629325305547115520033644135072042712842227470365931093763768283587123695713}{1852255105364486340941704390414913611729551082633346915338242324303666864557946209593538144799682} a^{3} + \frac{250216281851039386761078733509385247526872489829882257823992224458799055137247200208223024110971}{1190735424877169790605381393838158750397568553121437302717441494195214412930108277595845950228367} a^{2} + \frac{256639988224422448252837226119385558384980367801677229436575697460128272670679214057077318378637}{2778382658046729511412556585622370417594326623950020373007363486455500296836919314390307217199523} a - \frac{3785071772151188847377398261509778526628654367559977038736907664594825162367286175147166337382625}{8335147974140188534237669756867111252782979871850061119022090459366500890510757943170921651598569}$

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

3.1.21224.1, 7.1.4520453669548992.7

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $21$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.14.0.1}{14} }{,}\,{\href{/LocalNumberField/43.7.0.1}{7} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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
$2$2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.14.27.21$x^{14} + 4 x^{12} + 4 x^{11} - 4 x^{10} + 8 x^{9} + 2 x^{8} + 8 x^{6} + 4 x^{5} - 4 x^{4} + 6 x^{2} - 4 x - 6$$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$$[\ ]_{7}^{6}$
3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
$7$7.7.13.6$x^{7} + 203$$7$$1$$13$$F_7$$[13/6]_{6}$
7.14.27.56$x^{14} + 49 x^{13} - 147 x^{12} - 49 x^{11} + 147 x^{10} - 49 x^{9} + 98 x^{8} - 42 x^{7} - 147 x^{6} + 147 x^{5} + 49 x^{3} + 49 x^{2} + 49 x - 42$$14$$1$$27$$F_7$$[13/6]_{6}$
379Data not computed