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Properties

Label 21.1.30288687440...0000.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{18}\cdot 3^{19}\cdot 5^{19}\cdot 7^{15}\cdot 29^{7}\cdot 181^{7}$
Root discriminant $1464.79$
Ramified primes $2, 3, 5, 7, 29, 181$
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![-374806423239554574, 24293782425706416, -647042556248046, 8108854969694472, -449969373535374, 11228678522208, -75107387510946, 3472200615087, -60913514214, 386285674383, -14287865130, 200655063, -1191655710, 33070275, -306090, 2205441, -40824, 189, -2268, 21, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 + 21*x^19 - 2268*x^18 + 189*x^17 - 40824*x^16 + 2205441*x^15 - 306090*x^14 + 33070275*x^13 - 1191655710*x^12 + 200655063*x^11 - 14287865130*x^10 + 386285674383*x^9 - 60913514214*x^8 + 3472200615087*x^7 - 75107387510946*x^6 + 11228678522208*x^5 - 449969373535374*x^4 + 8108854969694472*x^3 - 647042556248046*x^2 + 24293782425706416*x - 374806423239554574)
 
gp: K = bnfinit(x^21 + 21*x^19 - 2268*x^18 + 189*x^17 - 40824*x^16 + 2205441*x^15 - 306090*x^14 + 33070275*x^13 - 1191655710*x^12 + 200655063*x^11 - 14287865130*x^10 + 386285674383*x^9 - 60913514214*x^8 + 3472200615087*x^7 - 75107387510946*x^6 + 11228678522208*x^5 - 449969373535374*x^4 + 8108854969694472*x^3 - 647042556248046*x^2 + 24293782425706416*x - 374806423239554574, 1)
 

Normalized defining polynomial

\( x^{21} + 21 x^{19} - 2268 x^{18} + 189 x^{17} - 40824 x^{16} + 2205441 x^{15} - 306090 x^{14} + 33070275 x^{13} - 1191655710 x^{12} + 200655063 x^{11} - 14287865130 x^{10} + 386285674383 x^{9} - 60913514214 x^{8} + 3472200615087 x^{7} - 75107387510946 x^{6} + 11228678522208 x^{5} - 449969373535374 x^{4} + 8108854969694472 x^{3} - 647042556248046 x^{2} + 24293782425706416 x - 374806423239554574 \)

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:  \(3028868744050989768689551478105098411269081489345000000000000000000=2^{18}\cdot 3^{19}\cdot 5^{19}\cdot 7^{15}\cdot 29^{7}\cdot 181^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1464.79$
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:  $2, 3, 5, 7, 29, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{3} a^{8}$, $\frac{1}{3} a^{9}$, $\frac{1}{3} a^{10}$, $\frac{1}{3} a^{11}$, $\frac{1}{9} a^{12} + \frac{1}{3} a^{5}$, $\frac{1}{9} a^{13} + \frac{1}{3} a^{6}$, $\frac{1}{90} a^{14} - \frac{1}{90} a^{13} + \frac{1}{30} a^{12} + \frac{1}{30} a^{11} + \frac{1}{10} a^{10} - \frac{1}{30} a^{9} + \frac{1}{30} a^{8} - \frac{1}{10} a^{7} + \frac{1}{15} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{180} a^{15} - \frac{2}{45} a^{13} - \frac{1}{45} a^{12} - \frac{1}{10} a^{11} - \frac{2}{15} a^{10} + \frac{2}{15} a^{8} + \frac{3}{20} a^{7} + \frac{7}{15} a^{6} + \frac{1}{3} a^{5} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{1}{10}$, $\frac{1}{540} a^{16} - \frac{1}{45} a^{13} + \frac{1}{90} a^{12} + \frac{2}{15} a^{10} + \frac{1}{9} a^{9} - \frac{1}{60} a^{8} + \frac{2}{15} a^{7} - \frac{7}{15} a^{6} - \frac{1}{15} a^{5} + \frac{2}{5} a^{3} + \frac{7}{15} a^{2} - \frac{1}{10} a - \frac{2}{5}$, $\frac{1}{540} a^{17} - \frac{1}{90} a^{13} - \frac{2}{45} a^{12} - \frac{2}{15} a^{11} - \frac{1}{45} a^{10} - \frac{1}{12} a^{9} - \frac{2}{15} a^{8} + \frac{1}{15} a^{6} + \frac{1}{15} a^{5} - \frac{2}{15} a^{3} - \frac{3}{10} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{3780} a^{18} + \frac{1}{3780} a^{17} - \frac{1}{630} a^{15} - \frac{1}{210} a^{14} + \frac{1}{126} a^{13} - \frac{4}{315} a^{12} - \frac{16}{315} a^{11} - \frac{1}{180} a^{10} - \frac{29}{420} a^{9} + \frac{13}{105} a^{8} + \frac{3}{70} a^{7} - \frac{44}{105} a^{6} - \frac{8}{21} a^{5} + \frac{7}{15} a^{4} - \frac{103}{210} a^{3} + \frac{3}{14} a^{2} - \frac{8}{35} a - \frac{16}{35}$, $\frac{1}{7125300} a^{19} - \frac{701}{7125300} a^{18} - \frac{1717}{7125300} a^{17} + \frac{4663}{7125300} a^{16} - \frac{89}{182700} a^{15} + \frac{971}{237510} a^{14} + \frac{1403}{79170} a^{13} - \frac{257}{6090} a^{12} - \frac{45893}{475020} a^{11} - \frac{46321}{475020} a^{10} - \frac{206149}{2375100} a^{9} - \frac{92647}{791700} a^{8} - \frac{26959}{791700} a^{7} + \frac{27813}{65975} a^{6} + \frac{92522}{197925} a^{5} - \frac{190363}{395850} a^{4} + \frac{36343}{395850} a^{3} - \frac{166969}{395850} a^{2} + \frac{5081}{18850} a - \frac{391}{4550}$, $\frac{1}{410974478311677630587257316932545904985173469789536717403698931498662003263006590459183617208457679301461110503554369218900} a^{20} + \frac{3944418203597301627083460112512706771487944869400769188616762609869176428309386373118424522340767510130759089280}{65234044176456766759882113798816810315106899966593129746618878015660635438572474676060891620390107825628747698976884003} a^{19} - \frac{20285207914657536751423107504957800674920353525769278714082745684434051792405196397793592954617304887814959849968531239}{205487239155838815293628658466272952492586734894768358701849465749331001631503295229591808604228839650730555251777184609450} a^{18} - \frac{31252776643373222574487528688523594247076635192491234382636131296752055595030370181429825708811969476662643742766187096}{102743619577919407646814329233136476246293367447384179350924732874665500815751647614795904302114419825365277625888592304725} a^{17} + \frac{221039940761479852663097952790509419370647778821873528022634365872694600091243887894449006737914215479987961124997255697}{410974478311677630587257316932545904985173469789536717403698931498662003263006590459183617208457679301461110503554369218900} a^{16} + \frac{915375153213454699490140010436481607302085852381049363458598493776271595194607774855428261228065330533352020184270107}{15221276974506578910639159886390589073524943325538396940877738203654148269000244091080874711424358492646707796427939600700} a^{15} - \frac{9747060889757456750131385163181053726510502411851190045812428421409951757717007869600582823390891621864988156395443039}{1957021325293703002796463413964504309453206998997793892398566340469819063157174240281826748611703234768862430969306520090} a^{14} - \frac{19004423217800805198265696576546833013356752650591874506447988537916596280468544024385884742657166183111821900376194783}{2739829855411184203915048779550306033234489798596911449357992876657746688420043936394557448056384528676407403357029128126} a^{13} - \frac{1018379526136067207255869254805718344320223371245890648811335847623445584029073615712256625214312169830133084189476114059}{27398298554111842039150487795503060332344897985969114493579928766577466884200439363945574480563845286764074033570291281260} a^{12} + \frac{766561091477962476559624946741807941829899397663807925123548876288874721091754656911620898545123935001613595514707594578}{6849574638527960509787621948875765083086224496492278623394982191644366721050109840986393620140961321691018508392572820315} a^{11} + \frac{265368693749182151978365346430674937440826293096853105332986634610053353280986091622880841972576329739023270319807419874}{34247873192639802548938109744378825415431122482461393116974910958221833605250549204931968100704806608455092541962864101575} a^{10} - \frac{308091987176383716770290663309402174316049005836004740852432263109683264735238181627191066535551398547221353446959670441}{2739829855411184203915048779550306033234489798596911449357992876657746688420043936394557448056384528676407403357029128126} a^{9} + \frac{2451831648270837673560031623515797223859457619071883146072401625737392928500606420630763829719608771786527219583229577053}{15221276974506578910639159886390589073524943325538396940877738203654148269000244091080874711424358492646707796427939600700} a^{8} - \frac{3864075858823017107948016119503837847002472287554750381054462474537025097346730929003568202496458697242862443738508685663}{45663830923519736731917479659171767220574829976615190822633214610962444807000732273242624134273075477940123389283818802100} a^{7} + \frac{3264922123371849968365278960243056749609269057838521985768719797072182801758942081329637089769331548540605439937536217906}{11415957730879934182979369914792941805143707494153797705658303652740611201750183068310656033568268869485030847320954700525} a^{6} + \frac{7943623456934327296172934739795354686912295103058881326756263347593843675320440753719986367113280719479506505724600143411}{22831915461759868365958739829585883610287414988307595411316607305481222403500366136621312067136537738970061694641909401050} a^{5} - \frac{377860399411603612834830259643468100225344335491434126743381828740355372174043182723916493226524267979296057246476813008}{2283191546175986836595873982958588361028741498830759541131660730548122240350036613662131206713653773897006169464190940105} a^{4} + \frac{1826219434414817104553935129182456760674236396912686320669702208031633786746726096128592830126399203568589251430771715917}{11415957730879934182979369914792941805143707494153797705658303652740611201750183068310656033568268869485030847320954700525} a^{3} - \frac{1532863359436770759746295755502519026178566465508898183651376131445378356720259612465011672951111610114532723152031352264}{11415957730879934182979369914792941805143707494153797705658303652740611201750183068310656033568268869485030847320954700525} a^{2} + \frac{1091295318670463676189644820907778120904957281158732117224270286193099307262526151345491256620722262387769662034334609243}{7610638487253289455319579943195294536762471662769198470438869101827074134500122045540437355712179246323353898213969800350} a - \frac{50397846946033161811328444660590905689991415498112720827005834905511180325806248572224890797960575423136279452584016461}{262435809905285843286882067006734294371119712509282705877202382821623246017245587777256460541799284355977720628067924150}$

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

3.1.78735.1, 7.1.12252303000000.6

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 ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$[\ ]_{7}^{3}$
2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
2.7.6.1$x^{7} - 2$$7$$1$$6$$C_7:C_3$$[\ ]_{7}^{3}$
$3$3.7.6.1$x^{7} - 3$$7$$1$$6$$F_7$$[\ ]_{7}^{6}$
3.14.13.2$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$$[\ ]_{14}^{6}$
7Data not computed
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.4.2.1$x^{4} + 6335 x^{2} + 10614564$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
181.4.2.1$x^{4} + 6335 x^{2} + 10614564$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
181.4.2.1$x^{4} + 6335 x^{2} + 10614564$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$