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Properties

Label 21.21.1592187855...4009.1
Degree $21$
Signature $[21, 0]$
Discriminant $877^{14}$
Root discriminant $91.62$
Ramified prime $877$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_7:C_3$ (as 21T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1593, -337401, -3047523, 6245027, 49086315, -87722101, -10352270, 85473488, -22553276, -32356568, 14095477, 5652363, -3420406, -412080, 409371, 779, -25320, 1466, 768, -70, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 9*x^20 - 70*x^19 + 768*x^18 + 1466*x^17 - 25320*x^16 + 779*x^15 + 409371*x^14 - 412080*x^13 - 3420406*x^12 + 5652363*x^11 + 14095477*x^10 - 32356568*x^9 - 22553276*x^8 + 85473488*x^7 - 10352270*x^6 - 87722101*x^5 + 49086315*x^4 + 6245027*x^3 - 3047523*x^2 - 337401*x + 1593)
 
gp: K = bnfinit(x^21 - 9*x^20 - 70*x^19 + 768*x^18 + 1466*x^17 - 25320*x^16 + 779*x^15 + 409371*x^14 - 412080*x^13 - 3420406*x^12 + 5652363*x^11 + 14095477*x^10 - 32356568*x^9 - 22553276*x^8 + 85473488*x^7 - 10352270*x^6 - 87722101*x^5 + 49086315*x^4 + 6245027*x^3 - 3047523*x^2 - 337401*x + 1593, 1)
 

Normalized defining polynomial

\( x^{21} - 9 x^{20} - 70 x^{19} + 768 x^{18} + 1466 x^{17} - 25320 x^{16} + 779 x^{15} + 409371 x^{14} - 412080 x^{13} - 3420406 x^{12} + 5652363 x^{11} + 14095477 x^{10} - 32356568 x^{9} - 22553276 x^{8} + 85473488 x^{7} - 10352270 x^{6} - 87722101 x^{5} + 49086315 x^{4} + 6245027 x^{3} - 3047523 x^{2} - 337401 x + 1593 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[21, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(159218785599036824660651669785398798634009=877^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $91.62$
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:  $877$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} + \frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{9} - \frac{1}{12} a^{8} - \frac{1}{4} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} + \frac{1}{12} a^{4} - \frac{1}{3} a^{3} + \frac{1}{12} a^{2} - \frac{1}{12} a + \frac{1}{4}$, $\frac{1}{24} a^{12} - \frac{1}{24} a^{10} - \frac{1}{24} a^{9} - \frac{1}{12} a^{7} - \frac{1}{12} a^{6} + \frac{1}{24} a^{5} - \frac{1}{24} a^{4} - \frac{5}{24} a^{3} - \frac{1}{6} a^{2} + \frac{1}{8} a + \frac{1}{8}$, $\frac{1}{24} a^{13} - \frac{1}{24} a^{11} - \frac{1}{24} a^{10} - \frac{1}{12} a^{8} - \frac{1}{12} a^{7} + \frac{1}{24} a^{6} - \frac{1}{24} a^{5} - \frac{5}{24} a^{4} - \frac{1}{6} a^{3} + \frac{1}{8} a^{2} + \frac{1}{8} a$, $\frac{1}{24} a^{14} - \frac{1}{24} a^{11} - \frac{1}{24} a^{10} - \frac{1}{8} a^{9} - \frac{1}{12} a^{8} - \frac{1}{24} a^{7} - \frac{1}{8} a^{6} - \frac{1}{6} a^{5} - \frac{5}{24} a^{4} - \frac{1}{12} a^{3} - \frac{1}{24} a^{2} + \frac{1}{8} a + \frac{1}{8}$, $\frac{1}{42096} a^{15} - \frac{719}{42096} a^{14} - \frac{725}{42096} a^{13} + \frac{439}{42096} a^{12} - \frac{453}{14032} a^{11} - \frac{553}{42096} a^{10} + \frac{299}{42096} a^{9} + \frac{1981}{42096} a^{8} - \frac{571}{10524} a^{7} + \frac{103}{21048} a^{6} - \frac{3919}{21048} a^{5} + \frac{3017}{21048} a^{4} + \frac{10747}{42096} a^{3} - \frac{4599}{14032} a^{2} + \frac{2293}{14032} a + \frac{4355}{14032}$, $\frac{1}{126288} a^{16} - \frac{16}{7893} a^{14} - \frac{413}{31572} a^{13} + \frac{79}{31572} a^{12} - \frac{1051}{31572} a^{11} + \frac{382}{7893} a^{10} - \frac{1843}{21048} a^{9} + \frac{11839}{126288} a^{8} + \frac{7393}{31572} a^{7} - \frac{899}{63144} a^{6} - \frac{9221}{63144} a^{5} + \frac{27337}{126288} a^{4} + \frac{7495}{63144} a^{3} + \frac{11623}{63144} a^{2} - \frac{459}{7016} a - \frac{2805}{14032}$, $\frac{1}{126288} a^{17} - \frac{1}{126288} a^{15} - \frac{827}{126288} a^{14} - \frac{389}{126288} a^{13} + \frac{2501}{126288} a^{12} - \frac{3665}{126288} a^{11} - \frac{1579}{42096} a^{10} + \frac{4577}{63144} a^{9} + \frac{8527}{126288} a^{8} - \frac{2665}{31572} a^{7} + \frac{815}{7893} a^{6} + \frac{12421}{126288} a^{5} + \frac{4289}{31572} a^{4} - \frac{4081}{126288} a^{3} - \frac{3827}{42096} a^{2} - \frac{1507}{10524} a + \frac{3589}{14032}$, $\frac{1}{221509152} a^{18} - \frac{91}{55377288} a^{17} - \frac{133}{110754576} a^{16} - \frac{1141}{221509152} a^{15} - \frac{2259143}{221509152} a^{14} - \frac{349633}{24612128} a^{13} - \frac{2063663}{221509152} a^{12} - \frac{169349}{73836384} a^{11} + \frac{75549}{6153032} a^{10} + \frac{741517}{73836384} a^{9} + \frac{811477}{73836384} a^{8} - \frac{1389893}{27688644} a^{7} - \frac{2814397}{24612128} a^{6} + \frac{15824695}{110754576} a^{5} + \frac{24468991}{110754576} a^{4} - \frac{6385561}{221509152} a^{3} + \frac{4836367}{13844322} a^{2} - \frac{31044329}{73836384} a - \frac{3673149}{24612128}$, $\frac{1}{221509152} a^{19} + \frac{271}{110754576} a^{17} + \frac{259}{221509152} a^{16} + \frac{2137}{221509152} a^{15} - \frac{42349}{221509152} a^{14} - \frac{2292311}{221509152} a^{13} - \frac{939365}{73836384} a^{12} - \frac{46162}{6922161} a^{11} + \frac{25603279}{221509152} a^{10} + \frac{23593643}{221509152} a^{9} - \frac{1076609}{27688644} a^{8} + \frac{28859119}{221509152} a^{7} + \frac{24862843}{110754576} a^{6} - \frac{25909037}{110754576} a^{5} - \frac{1947285}{24612128} a^{4} + \frac{290792}{2307387} a^{3} - \frac{25064521}{73836384} a^{2} + \frac{35569021}{73836384} a + \frac{1715125}{6153032}$, $\frac{1}{1089982139799637380785465895264} a^{20} + \frac{110564806905067568799}{121109126644404153420607321696} a^{19} - \frac{2246558705324003205209}{1089982139799637380785465895264} a^{18} + \frac{1653396139778961327202897}{1089982139799637380785465895264} a^{17} + \frac{5093269996405043643365}{1401005321079225425174120688} a^{16} + \frac{9118989514384017405174625}{1089982139799637380785465895264} a^{15} - \frac{806217895433599675332236551}{363327379933212460261821965088} a^{14} + \frac{5159472284554922000109488467}{1089982139799637380785465895264} a^{13} - \frac{325204145767148690941337835}{30277281661101038355151830424} a^{12} + \frac{930940172023380842711811013}{272495534949909345196366473816} a^{11} + \frac{30140053527053407353399983461}{272495534949909345196366473816} a^{10} - \frac{20278730572851487868357199557}{272495534949909345196366473816} a^{9} - \frac{3877032340590796496310419995}{272495534949909345196366473816} a^{8} + \frac{84564827469087849721232716405}{363327379933212460261821965088} a^{7} + \frac{223815039758363884855964133943}{1089982139799637380785465895264} a^{6} + \frac{51543819947717462301142965199}{1089982139799637380785465895264} a^{5} - \frac{20711810550409395577545983399}{1089982139799637380785465895264} a^{4} - \frac{34146869251193677533204962477}{90831844983303115065455491272} a^{3} - \frac{83581667285958371289466667489}{181663689966606230130910982544} a^{2} + \frac{3350377740553228649633286037}{60554563322202076710303660848} a - \frac{806404164375139161256229081}{2052697061769561922383174944}$

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:  $20$
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:  \( 199665489400000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_7:C_3$ (as 21T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 21
The 5 conjugacy class representatives for $C_7:C_3$
Character table for $C_7:C_3$

Intermediate fields

3.3.769129.1, 7.7.591559418641.1 x7

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 7 sibling: 7.7.591559418641.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/11.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/17.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/29.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/43.7.0.1}{7} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{7}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{21}$

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
877Data not computed