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Properties

Label 21.1.93884104902...3168.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{18}\cdot 3^{18}\cdot 7^{17}\cdot 13^{19}\cdot 43^{7}$
Root discriminant $800.68$
Ramified primes $2, 3, 7, 13, 43$
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![22093742653584, 5617933981296, 2779476929040, 4969555055472, -1876730026416, 1425798699360, -511010249400, 196394111312, -56921931136, 17184792352, -4567524416, 1072325800, -220547236, 40655986, -6775769, 1077041, -132097, 18977, -1463, 203, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 203*x^19 - 1463*x^18 + 18977*x^17 - 132097*x^16 + 1077041*x^15 - 6775769*x^14 + 40655986*x^13 - 220547236*x^12 + 1072325800*x^11 - 4567524416*x^10 + 17184792352*x^9 - 56921931136*x^8 + 196394111312*x^7 - 511010249400*x^6 + 1425798699360*x^5 - 1876730026416*x^4 + 4969555055472*x^3 + 2779476929040*x^2 + 5617933981296*x + 22093742653584)
 
gp: K = bnfinit(x^21 - 7*x^20 + 203*x^19 - 1463*x^18 + 18977*x^17 - 132097*x^16 + 1077041*x^15 - 6775769*x^14 + 40655986*x^13 - 220547236*x^12 + 1072325800*x^11 - 4567524416*x^10 + 17184792352*x^9 - 56921931136*x^8 + 196394111312*x^7 - 511010249400*x^6 + 1425798699360*x^5 - 1876730026416*x^4 + 4969555055472*x^3 + 2779476929040*x^2 + 5617933981296*x + 22093742653584, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 203 x^{19} - 1463 x^{18} + 18977 x^{17} - 132097 x^{16} + 1077041 x^{15} - 6775769 x^{14} + 40655986 x^{13} - 220547236 x^{12} + 1072325800 x^{11} - 4567524416 x^{10} + 17184792352 x^{9} - 56921931136 x^{8} + 196394111312 x^{7} - 511010249400 x^{6} + 1425798699360 x^{5} - 1876730026416 x^{4} + 4969555055472 x^{3} + 2779476929040 x^{2} + 5617933981296 x + 22093742653584 \)

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:  \(9388410490242195341432275672675535918039385773886516991623168=2^{18}\cdot 3^{18}\cdot 7^{17}\cdot 13^{19}\cdot 43^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $800.68$
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:  $2, 3, 7, 13, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6}$, $\frac{1}{7} a^{7}$, $\frac{1}{182} a^{8} - \frac{5}{91} a^{7} - \frac{3}{91} a^{6} - \frac{6}{13} a^{5} + \frac{5}{26} a^{4} - \frac{3}{13} a^{3} + \frac{5}{13} a^{2} + \frac{2}{13} a - \frac{4}{13}$, $\frac{1}{182} a^{9} - \frac{1}{91} a^{7} + \frac{6}{91} a^{6} - \frac{11}{26} a^{5} - \frac{4}{13} a^{4} + \frac{1}{13} a^{3} + \frac{3}{13} a - \frac{1}{13}$, $\frac{1}{182} a^{10} - \frac{4}{91} a^{7} - \frac{11}{182} a^{6} - \frac{3}{13} a^{5} + \frac{6}{13} a^{4} - \frac{6}{13} a^{3} + \frac{3}{13} a + \frac{5}{13}$, $\frac{1}{1274} a^{11} - \frac{1}{14} a^{7} + \frac{1}{91} a^{6} - \frac{3}{91} a^{5} + \frac{2}{13} a^{4} + \frac{4}{13} a^{3} - \frac{5}{13} a^{2} + \frac{3}{13} a + \frac{1}{13}$, $\frac{1}{1274} a^{12} + \frac{1}{91} a^{7} - \frac{3}{91} a^{6} + \frac{2}{13} a^{5} - \frac{5}{26} a^{4} - \frac{5}{13} a^{3} + \frac{3}{13} a^{2} + \frac{1}{13} a$, $\frac{1}{1274} a^{13} - \frac{6}{91} a^{7} - \frac{6}{91} a^{6} - \frac{7}{26} a^{5} + \frac{3}{13} a^{4} - \frac{4}{13} a^{3} + \frac{4}{13} a^{2} - \frac{4}{13} a - \frac{5}{13}$, $\frac{1}{33124} a^{14} - \frac{11}{33124} a^{13} - \frac{5}{33124} a^{12} - \frac{3}{33124} a^{11} - \frac{1}{364} a^{10} + \frac{9}{4732} a^{9} - \frac{1}{4732} a^{8} - \frac{187}{4732} a^{7} - \frac{23}{2366} a^{6} + \frac{575}{1183} a^{5} - \frac{4}{13} a^{4} - \frac{14}{169} a^{3} + \frac{36}{169} a^{2} + \frac{50}{169} a + \frac{71}{169}$, $\frac{1}{66248} a^{15} - \frac{1}{66248} a^{14} - \frac{11}{66248} a^{13} - \frac{1}{66248} a^{12} - \frac{17}{66248} a^{11} - \frac{17}{9464} a^{10} - \frac{15}{9464} a^{9} + \frac{11}{9464} a^{8} + \frac{40}{1183} a^{7} - \frac{8}{169} a^{6} - \frac{321}{2366} a^{5} + \frac{103}{338} a^{4} - \frac{5}{26} a^{3} + \frac{111}{338} a^{2} + \frac{77}{338} a + \frac{30}{169}$, $\frac{1}{463736} a^{16} + \frac{3}{8281} a^{13} - \frac{1}{2548} a^{12} + \frac{5}{16562} a^{11} + \frac{9}{8281} a^{10} + \frac{1}{9464} a^{8} - \frac{19}{338} a^{7} - \frac{11}{169} a^{6} - \frac{435}{1183} a^{5} - \frac{384}{1183} a^{4} + \frac{8}{169} a^{3} - \frac{82}{169} a^{2} + \frac{161}{338} a + \frac{67}{169}$, $\frac{1}{6028568} a^{17} + \frac{1}{430612} a^{15} + \frac{3}{215306} a^{14} + \frac{57}{430612} a^{13} - \frac{9}{215306} a^{12} - \frac{61}{215306} a^{11} + \frac{25}{15379} a^{10} + \frac{37}{123032} a^{9} + \frac{5}{2197} a^{8} - \frac{1237}{61516} a^{7} - \frac{95}{30758} a^{6} - \frac{348}{2197} a^{5} + \frac{1951}{4394} a^{4} + \frac{170}{2197} a^{3} + \frac{673}{4394} a^{2} + \frac{167}{2197} a + \frac{563}{2197}$, $\frac{1}{6028568} a^{18} + \frac{1}{6028568} a^{16} - \frac{1}{861224} a^{15} - \frac{3}{861224} a^{14} - \frac{127}{861224} a^{13} + \frac{81}{861224} a^{12} - \frac{277}{861224} a^{11} - \frac{187}{107653} a^{10} - \frac{19}{123032} a^{9} + \frac{51}{30758} a^{8} + \frac{1513}{61516} a^{7} - \frac{551}{15379} a^{6} + \frac{13033}{30758} a^{5} - \frac{279}{15379} a^{4} + \frac{720}{2197} a^{3} + \frac{61}{4394} a^{2} + \frac{160}{2197} a + \frac{68}{169}$, $\frac{1}{12057136} a^{19} - \frac{1}{12057136} a^{18} - \frac{1}{12057136} a^{17} + \frac{5}{12057136} a^{16} + \frac{1}{246064} a^{15} + \frac{3}{246064} a^{14} + \frac{25}{246064} a^{13} - \frac{15}{132496} a^{12} + \frac{57}{861224} a^{11} - \frac{2207}{861224} a^{10} + \frac{15}{15379} a^{9} + \frac{1}{1183} a^{8} - \frac{355}{61516} a^{7} + \frac{505}{8788} a^{6} + \frac{22959}{61516} a^{5} + \frac{4961}{30758} a^{4} + \frac{1941}{4394} a^{3} - \frac{496}{2197} a^{2} + \frac{35}{169} a - \frac{186}{2197}$, $\frac{1}{1665401214004459103028000935291757467132055160398603238909060692834115025887312} a^{20} + \frac{4532293730346125831914951028494861451212229869518308686189139698705561}{237914459143494157575428705041679638161722165771229034129865813262016432269616} a^{19} + \frac{48320365720480122362302898188826820635193239694746042432792804186455283}{1665401214004459103028000935291757467132055160398603238909060692834115025887312} a^{18} + \frac{134398641846484403754464196333387070884496127034070653131577682028102079}{1665401214004459103028000935291757467132055160398603238909060692834115025887312} a^{17} + \frac{1718041131985227676685727277814333079317042337321406218661429803136014099}{1665401214004459103028000935291757467132055160398603238909060692834115025887312} a^{16} + \frac{1213896593095426882728608815207412739927882610817867388680620414326353435}{237914459143494157575428705041679638161722165771229034129865813262016432269616} a^{15} - \frac{2832181465608326697230385990790868656791637091206868617212974755999354351}{237914459143494157575428705041679638161722165771229034129865813262016432269616} a^{14} + \frac{46369475151766550344279416128836831671650261018665952477422642683501072975}{237914459143494157575428705041679638161722165771229034129865813262016432269616} a^{13} + \frac{9952561912812631197461264886149774663443912990510121199727710990715882781}{29739307392936769696928588130209954770215270721403629266233226657752054033702} a^{12} - \frac{19501813028926282524513969012669905183418821305232850935104201723019018641}{59478614785873539393857176260419909540430541442807258532466453315504108067404} a^{11} + \frac{6269103331019421548215990866835052836455326263087549331097551789968320597}{118957229571747078787714352520839819080861082885614517064932906631008216134808} a^{10} - \frac{638210993421491865085043615550140643554216524299396773600477267990738907}{4248472484705252813846941161458564967173610103057661323747603808250293433386} a^{9} - \frac{2764442707879897722467989010503394319567760446621121847411644959452820028}{2124236242352626406923470580729282483586805051528830661873801904125146716693} a^{8} - \frac{509164960819414907824341571621134177220556843490703090709012619833238726293}{8496944969410505627693882322917129934347220206115322647495207616500586866772} a^{7} - \frac{109328204158393624952677331930444776095729794231305822630118818740024332811}{8496944969410505627693882322917129934347220206115322647495207616500586866772} a^{6} - \frac{736190913349280198577986813020157621009899109801173832080380007775753004262}{2124236242352626406923470580729282483586805051528830661873801904125146716693} a^{5} + \frac{1805726820036392122296775220769537094371466327887701898053079234066198393101}{4248472484705252813846941161458564967173610103057661323747603808250293433386} a^{4} + \frac{2756586268761788907654716296838318496082016736293356541874793852603982647}{606924640672178973406705880208366423881944300436808760535371972607184776198} a^{3} + \frac{144044857968524825540458157063908801208856539799326252209068152241771938949}{303462320336089486703352940104183211940972150218404380267685986303592388099} a^{2} + \frac{2284301728213594256460393461499998105620401156258765981560981228173726495}{7057263263629988062868673025678679347464468609730334424829906658223078793} a + \frac{2205666468962076876136282527978343461567095127817928543181098200086882748}{7057263263629988062868673025678679347464468609730334424829906658223078793}$

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

3.1.27391.1, 7.1.3784929689032128.1

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.14.0.1}{14} }{,}\,{\href{/LocalNumberField/29.7.0.1}{7} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ R ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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}$
3.14.12.1$x^{14} - 3 x^{7} + 18$$7$$2$$12$$F_7$$[\ ]_{7}^{6}$
7Data not computed
$13$13.7.6.1$x^{7} - 13$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$
13.14.13.1$x^{14} - 13$$14$$1$$13$$D_{14}$$[\ ]_{14}^{2}$
$43$$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
$\Q_{43}$$x + 9$$1$$1$$0$Trivial$[\ ]$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$
43.2.1.2$x^{2} + 387$$2$$1$$1$$C_2$$[\ ]_{2}$