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Properties

Label 21.7.18651305623...2423.1
Degree $21$
Signature $[7, 7]$
Discriminant $-\,3^{7}\cdot 7^{2}\cdot 13^{8}\cdot 109^{7}\cdot 34163789377^{2}$
Root discriminant $221.93$
Ramified primes $3, 7, 13, 109, 34163789377$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 21T135

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5106939083179, 1904529690371, 2325188715868, -2348166641298, 1007663299963, -40805180781, -176322079186, 30556007833, 16592259769, -3609444791, -448788104, 230691434, -19252253, -7214026, 755118, 116779, -16412, -1320, 714, -1, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 6*x^20 - x^19 + 714*x^18 - 1320*x^17 - 16412*x^16 + 116779*x^15 + 755118*x^14 - 7214026*x^13 - 19252253*x^12 + 230691434*x^11 - 448788104*x^10 - 3609444791*x^9 + 16592259769*x^8 + 30556007833*x^7 - 176322079186*x^6 - 40805180781*x^5 + 1007663299963*x^4 - 2348166641298*x^3 + 2325188715868*x^2 + 1904529690371*x - 5106939083179)
 
gp: K = bnfinit(x^21 - 6*x^20 - x^19 + 714*x^18 - 1320*x^17 - 16412*x^16 + 116779*x^15 + 755118*x^14 - 7214026*x^13 - 19252253*x^12 + 230691434*x^11 - 448788104*x^10 - 3609444791*x^9 + 16592259769*x^8 + 30556007833*x^7 - 176322079186*x^6 - 40805180781*x^5 + 1007663299963*x^4 - 2348166641298*x^3 + 2325188715868*x^2 + 1904529690371*x - 5106939083179, 1)
 

Normalized defining polynomial

\( x^{21} - 6 x^{20} - x^{19} + 714 x^{18} - 1320 x^{17} - 16412 x^{16} + 116779 x^{15} + 755118 x^{14} - 7214026 x^{13} - 19252253 x^{12} + 230691434 x^{11} - 448788104 x^{10} - 3609444791 x^{9} + 16592259769 x^{8} + 30556007833 x^{7} - 176322079186 x^{6} - 40805180781 x^{5} + 1007663299963 x^{4} - 2348166641298 x^{3} + 2325188715868 x^{2} + 1904529690371 x - 5106939083179 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[7, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-18651305623818716641616831928805533457521463992423=-\,3^{7}\cdot 7^{2}\cdot 13^{8}\cdot 109^{7}\cdot 34163789377^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $221.93$
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:  $3, 7, 13, 109, 34163789377$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7} a^{16} + \frac{3}{7} a^{15} + \frac{3}{7} a^{14} - \frac{3}{7} a^{13} - \frac{2}{7} a^{12} - \frac{3}{7} a^{11} + \frac{3}{7} a^{10} - \frac{3}{7} a^{9} + \frac{2}{7} a^{7} + \frac{1}{7} a^{5} + \frac{3}{7} a^{4} + \frac{2}{7} a^{3} + \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{91} a^{17} - \frac{4}{91} a^{16} + \frac{38}{91} a^{15} - \frac{31}{91} a^{14} + \frac{33}{91} a^{13} + \frac{32}{91} a^{12} - \frac{3}{7} a^{11} + \frac{18}{91} a^{10} - \frac{3}{13} a^{9} - \frac{33}{91} a^{8} - \frac{1}{13} a^{7} + \frac{36}{91} a^{6} + \frac{3}{91} a^{5} + \frac{2}{91} a^{4} - \frac{3}{13} a^{3} + \frac{44}{91} a^{2} + \frac{19}{91} a + \frac{1}{13}$, $\frac{1}{637} a^{18} - \frac{1}{637} a^{17} - \frac{1}{49} a^{16} - \frac{125}{637} a^{15} + \frac{5}{637} a^{14} - \frac{116}{637} a^{13} - \frac{47}{637} a^{12} - \frac{164}{637} a^{11} + \frac{27}{91} a^{10} - \frac{23}{91} a^{9} - \frac{197}{637} a^{8} + \frac{4}{91} a^{7} - \frac{71}{637} a^{6} + \frac{5}{13} a^{5} - \frac{223}{637} a^{4} + \frac{176}{637} a^{3} - \frac{31}{637} a^{2} - \frac{41}{91} a + \frac{8}{637}$, $\frac{1}{637} a^{19} - \frac{12}{637} a^{16} - \frac{316}{637} a^{15} + \frac{1}{637} a^{14} - \frac{19}{49} a^{13} - \frac{127}{637} a^{12} + \frac{207}{637} a^{11} + \frac{27}{91} a^{10} + \frac{76}{637} a^{9} + \frac{6}{637} a^{8} + \frac{223}{637} a^{7} + \frac{41}{637} a^{6} + \frac{246}{637} a^{5} - \frac{110}{637} a^{4} + \frac{215}{637} a^{3} + \frac{298}{637} a^{2} - \frac{22}{49} a - \frac{258}{637}$, $\frac{1}{73718066121021763684364215571810906311802600085753756221667355228328690593102515683024654029307664579694300463331542620818708083} a^{20} + \frac{52183718930052519938230797724393734609652965842811502990751638297362988081490300377381040328943782763120319695115283515539769}{73718066121021763684364215571810906311802600085753756221667355228328690593102515683024654029307664579694300463331542620818708083} a^{19} - \frac{24842545116238166386220164158457405995217138972500762131430991208395991217626727989405720656019648470334952334249042107139183}{73718066121021763684364215571810906311802600085753756221667355228328690593102515683024654029307664579694300463331542620818708083} a^{18} - \frac{361616677062971269144940308472617313069262466794908902008976997681570952155736828411770230610916909176301402159161337007736811}{73718066121021763684364215571810906311802600085753756221667355228328690593102515683024654029307664579694300463331542620818708083} a^{17} - \frac{1179177835189564990184388654560682600096461805138118565786464753742787595590967657964947046648188087823265339001949133470852472}{73718066121021763684364215571810906311802600085753756221667355228328690593102515683024654029307664579694300463331542620818708083} a^{16} + \frac{4184164158155363861446889229534659194782788240521919739087365045042259451550057246290233778729687242603846936420767181170830871}{10531152303003109097766316510258700901686085726536250888809622175475527227586073669003522004186809225670614351904506088688386869} a^{15} - \frac{2075432466941164416773540826530517577340136337008509391500159499975522419443974363842887339466948524587513553779509964668832764}{5670620470847827975720324274754685100907892314288750478589796556025283891777116591001896463792897275361100035640887893909131391} a^{14} + \frac{19202496249334771677041296392364956607572036461087247540059669909511974326526129679011439032986182812938969470845694711263377392}{73718066121021763684364215571810906311802600085753756221667355228328690593102515683024654029307664579694300463331542620818708083} a^{13} + \frac{16281149165163180021892347100631463775036768917421219902093507322686176325475004408033718588848736066324797460431374074004404626}{73718066121021763684364215571810906311802600085753756221667355228328690593102515683024654029307664579694300463331542620818708083} a^{12} - \frac{3462636853299407599851839452397764925016848001764401859288859076289220686978936005421934751178869683626321379449929964584990248}{73718066121021763684364215571810906311802600085753756221667355228328690593102515683024654029307664579694300463331542620818708083} a^{11} - \frac{32884257676171207372971015574237239351747493370786722577581351118130233447422212228482331003931965547159014341887176848239981138}{73718066121021763684364215571810906311802600085753756221667355228328690593102515683024654029307664579694300463331542620818708083} a^{10} - \frac{2035542673627569597994071116110421053332575832825128423019684036482392797331636519501535579516048643208243116096460513068703721}{73718066121021763684364215571810906311802600085753756221667355228328690593102515683024654029307664579694300463331542620818708083} a^{9} - \frac{8490791851340922653367049245585502968025055335947260682831388459238002258918725187318901853620106158960895689490073364476712285}{73718066121021763684364215571810906311802600085753756221667355228328690593102515683024654029307664579694300463331542620818708083} a^{8} + \frac{27499068559810972663018785170690606492665853481175973902323637987684152498951031388603181945281786442830742185567859123620408820}{73718066121021763684364215571810906311802600085753756221667355228328690593102515683024654029307664579694300463331542620818708083} a^{7} + \frac{23702245031123909803044894662674437609071646788951438225958352286483674956398543182360335891922547030185547577176760093338055323}{73718066121021763684364215571810906311802600085753756221667355228328690593102515683024654029307664579694300463331542620818708083} a^{6} - \frac{10233841110346985848184184040197924410161243060237882195950242852991400538421701584592525962552692805659086897133794238054058814}{73718066121021763684364215571810906311802600085753756221667355228328690593102515683024654029307664579694300463331542620818708083} a^{5} - \frac{19084141886345728236792490936016597224397138513361287615883636068357397316660548023942656538480710057594684227109222067028935552}{73718066121021763684364215571810906311802600085753756221667355228328690593102515683024654029307664579694300463331542620818708083} a^{4} + \frac{7412583539471707451281146251028909390378034985220024784587799093875481462776883116601849630502315389803490694649774929140946763}{73718066121021763684364215571810906311802600085753756221667355228328690593102515683024654029307664579694300463331542620818708083} a^{3} - \frac{320009566138873485180505812547171998597431798870222927167447533401244627020678235316737176564972070666836637459059878317307537}{5670620470847827975720324274754685100907892314288750478589796556025283891777116591001896463792897275361100035640887893909131391} a^{2} - \frac{23598802254442282842639239765679870065540862331384641414878545956331846319827206959835378947735273330395498648638692546893465527}{73718066121021763684364215571810906311802600085753756221667355228328690593102515683024654029307664579694300463331542620818708083} a + \frac{25708074475538640637754766691639818052437314326422470415518848435845344335240619118345266298808136692238572910376571878983835225}{73718066121021763684364215571810906311802600085753756221667355228328690593102515683024654029307664579694300463331542620818708083}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 57057824105600000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

21T135:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 5878656
The 84 conjugacy class representatives for t21n135 are not computed
Character table for t21n135 is not computed

Intermediate fields

7.3.2007889.2

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 $21$ R ${\href{/LocalNumberField/5.14.0.1}{14} }{,}\,{\href{/LocalNumberField/5.7.0.1}{7} }$ R ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R ${\href{/LocalNumberField/17.9.0.1}{9} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ $21$ $21$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$3$3.7.0.1$x^{7} + x^{2} - x + 1$$1$$7$$0$$C_7$$[\ ]^{7}$
3.14.7.2$x^{14} + 243 x^{4} - 729 x^{2} + 2187$$2$$7$$7$$C_{14}$$[\ ]_{2}^{7}$
$7$7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.9.0.1$x^{9} + x^{2} - 6 x + 2$$1$$9$$0$$C_9$$[\ ]^{9}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.4.2.2$x^{4} - 13 x^{2} + 338$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
13.8.6.4$x^{8} - 13 x^{4} + 338$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$
$109$$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
109.2.1.2$x^{2} + 654$$2$$1$$1$$C_2$$[\ ]_{2}$
109.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
109.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
109.4.2.1$x^{4} + 1199 x^{2} + 427716$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
34163789377Data not computed