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Properties

Label 21.1.39885841885...5888.1
Degree $21$
Signature $[1, 10]$
Discriminant $2^{33}\cdot 3^{19}\cdot 7^{40}\cdot 13^{7}$
Root discriminant $768.69$
Ramified primes $2, 3, 7, 13$
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![14403261843, -4043917773, 7341777072, 211758120, 952919793, 815747877, -611271612, 513894856, -263747806, 134840062, -60738860, 24182648, -7649978, 2291226, -561156, 141288, -25445, 5327, -644, 112, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7*x^20 + 112*x^19 - 644*x^18 + 5327*x^17 - 25445*x^16 + 141288*x^15 - 561156*x^14 + 2291226*x^13 - 7649978*x^12 + 24182648*x^11 - 60738860*x^10 + 134840062*x^9 - 263747806*x^8 + 513894856*x^7 - 611271612*x^6 + 815747877*x^5 + 952919793*x^4 + 211758120*x^3 + 7341777072*x^2 - 4043917773*x + 14403261843)
 
gp: K = bnfinit(x^21 - 7*x^20 + 112*x^19 - 644*x^18 + 5327*x^17 - 25445*x^16 + 141288*x^15 - 561156*x^14 + 2291226*x^13 - 7649978*x^12 + 24182648*x^11 - 60738860*x^10 + 134840062*x^9 - 263747806*x^8 + 513894856*x^7 - 611271612*x^6 + 815747877*x^5 + 952919793*x^4 + 211758120*x^3 + 7341777072*x^2 - 4043917773*x + 14403261843, 1)
 

Normalized defining polynomial

\( x^{21} - 7 x^{20} + 112 x^{19} - 644 x^{18} + 5327 x^{17} - 25445 x^{16} + 141288 x^{15} - 561156 x^{14} + 2291226 x^{13} - 7649978 x^{12} + 24182648 x^{11} - 60738860 x^{10} + 134840062 x^{9} - 263747806 x^{8} + 513894856 x^{7} - 611271612 x^{6} + 815747877 x^{5} + 952919793 x^{4} + 211758120 x^{3} + 7341777072 x^{2} - 4043917773 x + 14403261843 \)

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:  \(3988584188566184927712460097535808514732885044093734132645888=2^{33}\cdot 3^{19}\cdot 7^{40}\cdot 13^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $768.69$
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:  $2, 3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{12} a^{8} + \frac{1}{12} a^{7} - \frac{1}{4} a^{6} + \frac{1}{12} a^{5} + \frac{1}{12} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{12} a^{9} - \frac{1}{12} a^{7} + \frac{1}{12} a^{6} - \frac{1}{4} a^{5} - \frac{1}{12} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{24} a^{10} - \frac{1}{24} a^{8} - \frac{1}{12} a^{7} + \frac{1}{12} a^{5} + \frac{1}{4} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a - \frac{1}{8}$, $\frac{1}{24} a^{11} - \frac{1}{24} a^{9} + \frac{1}{12} a^{7} - \frac{1}{6} a^{6} + \frac{1}{12} a^{5} - \frac{1}{6} a^{4} + \frac{3}{8} a^{3} + \frac{1}{8} a$, $\frac{1}{24} a^{12} - \frac{1}{24} a^{8} - \frac{1}{12} a^{7} + \frac{1}{12} a^{6} + \frac{1}{12} a^{5} + \frac{1}{24} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{3}{8}$, $\frac{1}{24} a^{13} - \frac{1}{24} a^{9} - \frac{1}{12} a^{7} + \frac{1}{12} a^{6} - \frac{1}{8} a^{5} + \frac{1}{12} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{8} a + \frac{1}{4}$, $\frac{1}{7056} a^{14} - \frac{5}{252} a^{13} - \frac{1}{336} a^{12} - \frac{5}{504} a^{11} - \frac{19}{1008} a^{10} - \frac{1}{56} a^{9} - \frac{5}{1008} a^{8} + \frac{163}{1764} a^{7} - \frac{53}{336} a^{6} - \frac{1}{7} a^{5} + \frac{17}{336} a^{4} - \frac{13}{56} a^{3} + \frac{51}{112} a^{2} - \frac{5}{56} a + \frac{235}{784}$, $\frac{1}{7056} a^{15} + \frac{11}{1008} a^{13} - \frac{5}{504} a^{12} + \frac{1}{112} a^{11} + \frac{5}{504} a^{10} + \frac{37}{1008} a^{9} - \frac{11}{588} a^{8} + \frac{29}{1008} a^{7} + \frac{1}{42} a^{6} - \frac{53}{336} a^{5} - \frac{25}{168} a^{4} - \frac{5}{112} a^{3} + \frac{23}{56} a^{2} - \frac{255}{784} a - \frac{1}{28}$, $\frac{1}{14112} a^{16} - \frac{1}{84} a^{13} - \frac{1}{168} a^{12} + \frac{1}{84} a^{11} - \frac{1}{168} a^{10} + \frac{19}{588} a^{9} + \frac{13}{336} a^{8} - \frac{1}{252} a^{7} - \frac{3}{14} a^{6} + \frac{1}{14} a^{5} + \frac{19}{168} a^{4} + \frac{1}{7} a^{3} + \frac{169}{392} a^{2} - \frac{1}{7} a + \frac{75}{224}$, $\frac{1}{42336} a^{17} + \frac{1}{42336} a^{16} - \frac{1}{168} a^{13} - \frac{1}{84} a^{12} + \frac{1}{63} a^{11} - \frac{67}{3528} a^{10} + \frac{23}{2352} a^{9} + \frac{11}{432} a^{8} - \frac{5}{378} a^{7} - \frac{11}{84} a^{6} + \frac{5}{56} a^{5} - \frac{5}{28} a^{4} + \frac{31}{98} a^{3} - \frac{191}{392} a^{2} + \frac{295}{672} a - \frac{137}{672}$, $\frac{1}{42336} a^{18} - \frac{1}{42336} a^{16} - \frac{1}{168} a^{13} - \frac{1}{72} a^{12} + \frac{1}{147} a^{11} - \frac{13}{1008} a^{10} - \frac{275}{10584} a^{9} + \frac{1}{336} a^{8} - \frac{53}{756} a^{7} - \frac{13}{84} a^{6} - \frac{17}{168} a^{5} + \frac{47}{392} a^{4} + \frac{9}{28} a^{3} - \frac{2111}{4704} a^{2} - \frac{15}{56} a - \frac{139}{672}$, $\frac{1}{296352} a^{19} - \frac{1}{98784} a^{18} - \frac{1}{148176} a^{17} - \frac{5}{148176} a^{16} - \frac{1}{16464} a^{15} - \frac{1}{16464} a^{14} + \frac{127}{7056} a^{13} - \frac{299}{16464} a^{12} + \frac{101}{6174} a^{11} - \frac{11}{74088} a^{10} - \frac{1985}{49392} a^{9} - \frac{1339}{148176} a^{8} - \frac{10187}{148176} a^{7} + \frac{575}{2352} a^{6} - \frac{3155}{16464} a^{5} - \frac{2785}{16464} a^{4} + \frac{4255}{32928} a^{3} + \frac{2733}{10976} a^{2} - \frac{163}{4116} a + \frac{607}{4116}$, $\frac{1}{2453123133405044560498075177712662748228772420391606971100579008} a^{20} + \frac{831912887717954541969176861370396588357494181301530693823}{613280783351261140124518794428165687057193105097901742775144752} a^{19} - \frac{811398281196807025913073272650547475367184153874473870957}{175223080957503182892719655550904482016340887170829069364327072} a^{18} + \frac{1249390799482703034190531028438311883550064339374421638725}{408853855567507426749679196285443791371462070065267828516763168} a^{17} - \frac{21744760811941700720333486986004457626824530107865652322069}{2453123133405044560498075177712662748228772420391606971100579008} a^{16} + \frac{7832245928027511831470831255933048638190240273406751874875}{204426927783753713374839598142721895685731035032633914258381584} a^{15} + \frac{1908444433170434532599351134734600568861029162015627194713}{102213463891876856687419799071360947842865517516316957129190792} a^{14} - \frac{2639490531928594296645726319785534690724392205733972262314933}{204426927783753713374839598142721895685731035032633914258381584} a^{13} - \frac{1714399917064443095397096597955350999094404608323885166858735}{136284618522502475583226398761814597123820690021755942838921056} a^{12} + \frac{495605405075547728351383982153530638551473286255189003737333}{87611540478751591446359827775452241008170443585414534682163536} a^{11} - \frac{9310694971865099739017599341295756501944564531116248651632671}{613280783351261140124518794428165687057193105097901742775144752} a^{10} - \frac{3468593986621238041679507532873084681418198298547488675568427}{153320195837815285031129698607041421764298276274475435693786188} a^{9} + \frac{8501749627712672486099338771159819862601214721528611805327369}{408853855567507426749679196285443791371462070065267828516763168} a^{8} - \frac{75667387136808406089768137587946187749312941333900561871166905}{613280783351261140124518794428165687057193105097901742775144752} a^{7} + \frac{2108077591781759600923721161188643628500828919951981456872445}{34071154630625618895806599690453649280955172505438985709730264} a^{6} + \frac{2645968220983287167225180138019041502292156702849109127093393}{22714103087083745930537733126969099520636781670292657139820176} a^{5} - \frac{146973099350556924141251741588206636153921160549856584235851}{5562637490714386758499036684155697841788599592724732360772288} a^{4} + \frac{14581708721835648845597190305630228961526007482793444566119059}{34071154630625618895806599690453649280955172505438985709730264} a^{3} - \frac{601808287409043898824918600200001225673436185858993104177743}{136284618522502475583226398761814597123820690021755942838921056} a^{2} - \frac{4624641086930938562962789126102697378267542034238302000395837}{45428206174167491861075466253938199041273563340585314279640352} a - \frac{123089276031085556346704139102763738161017991500394723309305753}{272569237045004951166452797523629194247641380043511885677842112}$

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

3.1.2184.1, 7.1.4520453669548992.29

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 ${\href{/LocalNumberField/11.3.0.1}{3} }^{7}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ $21$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$[\ ]_{7}^{3}$
2.14.27.121$x^{14} - 4 x^{13} - 4 x^{12} + 8 x^{11} + 8 x^{10} - 6 x^{8} + 8 x^{7} + 6 x^{6} - 4 x^{5} - 2 x^{4} - 4 x^{3} + 4 x^{2} + 8 x + 6$$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$$[\ ]_{7}^{6}$
3.14.13.1$x^{14} - 3$$14$$1$$13$$F_7 \times C_2$$[\ ]_{14}^{6}$
$7$7.7.13.7$x^{7} + 252$$7$$1$$13$$F_7$$[13/6]_{6}$
7.14.27.51$x^{14} - 98 x^{12} - 147 x^{11} + 98 x^{8} + 77 x^{7} + 147 x^{6} + 98 x^{5} - 98 x^{4} - 98 x^{3} - 98 x^{2} - 147 x + 77$$14$$1$$27$$F_7$$[13/6]_{6}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$