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Properties

Label 21.1.13345529421...8688.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{18}\cdot 3^{19}\cdot 7^{40}\cdot 491^{7}$
Root discriminant $1571.97$
Ramified primes $2, 3, 7, 491$
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![29930107102308, -28321950476484, 9674836268076, 1407559447980, -2591283279564, 737941558788, 66246386724, -72186530568, 11623314444, 2516687964, -1367972340, 136557204, 39839016, -11687256, 325961, 316603, -46977, -1729, 1057, -63, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 - 63*x^19 + 1057*x^18 - 1729*x^17 - 46977*x^16 + 316603*x^15 + 325961*x^14 - 11687256*x^13 + 39839016*x^12 + 136557204*x^11 - 1367972340*x^10 + 2516687964*x^9 + 11623314444*x^8 - 72186530568*x^7 + 66246386724*x^6 + 737941558788*x^5 - 2591283279564*x^4 + 1407559447980*x^3 + 9674836268076*x^2 - 28321950476484*x + 29930107102308)
 
gp: K = bnfinit(x^21 - 7*x^20 - 63*x^19 + 1057*x^18 - 1729*x^17 - 46977*x^16 + 316603*x^15 + 325961*x^14 - 11687256*x^13 + 39839016*x^12 + 136557204*x^11 - 1367972340*x^10 + 2516687964*x^9 + 11623314444*x^8 - 72186530568*x^7 + 66246386724*x^6 + 737941558788*x^5 - 2591283279564*x^4 + 1407559447980*x^3 + 9674836268076*x^2 - 28321950476484*x + 29930107102308, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} - 63 x^{19} + 1057 x^{18} - 1729 x^{17} - 46977 x^{16} + 316603 x^{15} + 325961 x^{14} - 11687256 x^{13} + 39839016 x^{12} + 136557204 x^{11} - 1367972340 x^{10} + 2516687964 x^{9} + 11623314444 x^{8} - 72186530568 x^{7} + 66246386724 x^{6} + 737941558788 x^{5} - 2591283279564 x^{4} + 1407559447980 x^{3} + 9674836268076 x^{2} - 28321950476484 x + 29930107102308 \)

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:  \(13345529421114240689213311204151578981153204581639050292194699378688=2^{18}\cdot 3^{19}\cdot 7^{40}\cdot 491^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1571.97$
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:  $2, 3, 7, 491$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{7} a^{8} - \frac{2}{7} a^{7} - \frac{2}{7} a^{6} + \frac{3}{7} a^{5} - \frac{3}{7} a^{4}$, $\frac{1}{7} a^{9} + \frac{1}{7} a^{7} - \frac{1}{7} a^{6} + \frac{3}{7} a^{5} + \frac{1}{7} a^{4}$, $\frac{1}{7} a^{10} + \frac{1}{7} a^{7} - \frac{2}{7} a^{6} - \frac{2}{7} a^{5} + \frac{3}{7} a^{4}$, $\frac{1}{7} a^{11} + \frac{3}{7} a^{4}$, $\frac{1}{7} a^{12} + \frac{3}{7} a^{5}$, $\frac{1}{7} a^{13} + \frac{3}{7} a^{6}$, $\frac{1}{4116} a^{14} + \frac{17}{588} a^{13} - \frac{5}{196} a^{12} + \frac{37}{588} a^{11} - \frac{1}{588} a^{10} + \frac{5}{196} a^{9} - \frac{41}{588} a^{8} + \frac{1109}{4116} a^{7} + \frac{2}{7} a^{6} - \frac{1}{14} a^{5} - \frac{5}{14} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{3}{14}$, $\frac{1}{8232} a^{15} - \frac{1}{8232} a^{14} - \frac{13}{392} a^{13} + \frac{73}{1176} a^{12} + \frac{11}{1176} a^{11} - \frac{11}{392} a^{10} - \frac{11}{168} a^{9} - \frac{319}{8232} a^{8} + \frac{33}{686} a^{7} + \frac{13}{28} a^{6} - \frac{5}{28} a^{5} + \frac{5}{28} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{11}{28} a - \frac{1}{7}$, $\frac{1}{8232} a^{16} - \frac{1}{98} a^{13} + \frac{1}{196} a^{12} + \frac{3}{98} a^{11} - \frac{2}{49} a^{10} - \frac{13}{343} a^{9} + \frac{11}{392} a^{8} + \frac{41}{294} a^{7} + \frac{1}{7} a^{6} + \frac{5}{14} a^{5} - \frac{5}{14} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{57624} a^{17} + \frac{1}{28812} a^{16} - \frac{1}{57624} a^{15} - \frac{1}{57624} a^{14} + \frac{71}{8232} a^{13} + \frac{425}{8232} a^{12} + \frac{527}{8232} a^{11} - \frac{2503}{57624} a^{10} + \frac{719}{14406} a^{9} - \frac{2789}{57624} a^{8} + \frac{2733}{9604} a^{7} + \frac{81}{196} a^{6} - \frac{11}{28} a^{5} + \frac{1}{4} a^{4} - \frac{45}{196} a^{3} - \frac{17}{98} a^{2} - \frac{53}{196} a + \frac{33}{98}$, $\frac{1}{57624} a^{18} + \frac{1}{28812} a^{16} + \frac{1}{57624} a^{15} - \frac{5}{57624} a^{14} - \frac{137}{8232} a^{13} + \frac{223}{8232} a^{12} + \frac{1291}{57624} a^{11} + \frac{59}{4116} a^{10} + \frac{737}{19208} a^{9} + \frac{3601}{57624} a^{8} - \frac{53}{28812} a^{7} + \frac{97}{196} a^{6} - \frac{13}{28} a^{5} - \frac{59}{196} a^{4} + \frac{2}{7} a^{3} - \frac{55}{196} a^{2} - \frac{73}{196} a - \frac{45}{98}$, $\frac{1}{2247336} a^{19} - \frac{1}{561834} a^{18} + \frac{1}{374556} a^{17} + \frac{127}{2247336} a^{16} + \frac{113}{2247336} a^{15} - \frac{53}{749112} a^{14} + \frac{7747}{321048} a^{13} + \frac{30467}{2247336} a^{12} + \frac{2651}{374556} a^{11} + \frac{17009}{749112} a^{10} + \frac{44843}{749112} a^{9} + \frac{2929}{62426} a^{8} + \frac{73889}{187278} a^{7} - \frac{97}{588} a^{6} - \frac{263}{588} a^{5} - \frac{146}{637} a^{4} + \frac{631}{2548} a^{3} + \frac{1021}{2548} a^{2} - \frac{163}{637} a + \frac{55}{1274}$, $\frac{1}{899347433657839115845459452294390259488548556579828803035256551406539045553770117718199638808} a^{20} + \frac{33454925505658830933989085954081602419202729176096457175511075038152975018271132430403}{899347433657839115845459452294390259488548556579828803035256551406539045553770117718199638808} a^{19} + \frac{2320559979110334071968089485082320338986084553048421023274183179264065659987658836643965}{299782477885946371948486484098130086496182852193276267678418850468846348517923372572733212936} a^{18} - \frac{18687809933101447682585236590618658869323288296708598659873167186448740686464575645079}{8647571477479222267744802425907598648928351505575276952262082225062875438017020362674996527} a^{17} - \frac{761389561046780796941170441768544389824721771471509110956217356246006525342391195156053}{32119551202065682708766409010513937838876734163565314394116305407376394484063218489935701386} a^{16} + \frac{4952793698935062213634247404581646749871804309119044979814425586353397427199857010192347}{149891238942973185974243242049065043248091426096638133839209425234423174258961686286366606468} a^{15} - \frac{14777063985224720849345788667180537256937572926483642892634441164781823805696840113235635}{224836858414459778961364863073597564872137139144957200758814137851634761388442529429549909702} a^{14} + \frac{21805363995652653082220423794515255355193811934227035414529596596169779761512860304383571281}{449673716828919557922729726147195129744274278289914401517628275703269522776885058859099819404} a^{13} - \frac{6981714321406060097031951995343712449616988833037263606289097845020169589669976411919192597}{299782477885946371948486484098130086496182852193276267678418850468846348517923372572733212936} a^{12} + \frac{2133811994091265543374722556691205886829382561893942864756163328368230348227343057095098663}{99927492628648790649495494699376695498727617397758755892806283489615449505974457524244404312} a^{11} - \frac{16097905305204421272441595104902776401083209409302625229544530586139979834720253112669722475}{299782477885946371948486484098130086496182852193276267678418850468846348517923372572733212936} a^{10} - \frac{19352052853644606399234586455541912332274275653998194975302556877707528473592905945185347}{7137678044903485046392535335669763964194829814125625420914734534972532107569604108874600308} a^{9} - \frac{765970803325334358482133036786709334326347592929486906743679236668512349893870689205868022}{37472809735743296493560810512266260812022856524159533459802356308605793564740421571591651617} a^{8} - \frac{48090834294958249910664892854325947909600782112629521059196723076165570469138887013230502025}{149891238942973185974243242049065043248091426096638133839209425234423174258961686286366606468} a^{7} - \frac{485470018747425908089669599473634670446550000582198978911854885409373419782290411475449}{117654033707200302962514318719831274135079612320752067377715404422624155619279188607823082} a^{6} + \frac{296174455427716174202911313697871863617592201258773418247720607001312182518893724289301087}{1019668292129069292341790762238537709170689973446517917273533504996076015367086301267800044} a^{5} + \frac{169399429989033632334291640694787543797438176596924134978534793087716971958031053282830047}{1019668292129069292341790762238537709170689973446517917273533504996076015367086301267800044} a^{4} - \frac{233005413616069976962051243638909130293431084577615756291993631351089066791777897073483923}{1019668292129069292341790762238537709170689973446517917273533504996076015367086301267800044} a^{3} + \frac{9242110900223078172834102833964532226445429784252584054974668292975827350530401324867344}{36416724718895331869349670079947775327524641908804211331197625178431286263110225045278573} a^{2} + \frac{75667714658932850842248984938986941916283356968815897535561106087656687046273226414041679}{254917073032267323085447690559634427292672493361629479318383376249019003841771575316950011} a - \frac{154106814797947514269875833106717471431557939052812464732909438379315317936091347753270839}{509834146064534646170895381119268854585344986723258958636766752498038007683543150633900022}$

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

3.1.10311.1, 7.1.4520453669548992.28

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ $21$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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}$
3.14.13.1$x^{14} - 3$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$
$7$7.7.13.4$x^{7} + 154$$7$$1$$13$$F_7$$[13/6]_{6}$
7.14.27.2$x^{14} + 49 x^{13} + 147 x^{12} - 147 x^{11} + 147 x^{10} + 98 x^{9} + 49 x^{8} - 14 x^{7} - 147 x^{6} + 147 x^{4} + 147 x^{3} - 147 x^{2} - 98 x - 28$$14$$1$$27$$F_7 \times C_2$$[13/6]_{6}^{2}$
491Data not computed