Properties

Label 27.27.735...441.1
Degree $27$
Signature $[27, 0]$
Discriminant $7.353\times 10^{47}$
Root discriminant $59.27$
Ramified primes $3, 19$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_3\times C_9$ (as 27T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 51*x^25 - 4*x^24 + 1080*x^23 + 156*x^22 - 12356*x^21 - 2448*x^20 + 83283*x^19 + 19984*x^18 - 339003*x^17 - 91596*x^16 + 825846*x^15 + 238428*x^14 - 1168977*x^13 - 344712*x^12 + 930681*x^11 + 259620*x^10 - 414755*x^9 - 102465*x^8 + 101628*x^7 + 20920*x^6 - 12933*x^5 - 2085*x^4 + 755*x^3 + 90*x^2 - 15*x - 1)
 
gp: K = bnfinit(x^27 - 51*x^25 - 4*x^24 + 1080*x^23 + 156*x^22 - 12356*x^21 - 2448*x^20 + 83283*x^19 + 19984*x^18 - 339003*x^17 - 91596*x^16 + 825846*x^15 + 238428*x^14 - 1168977*x^13 - 344712*x^12 + 930681*x^11 + 259620*x^10 - 414755*x^9 - 102465*x^8 + 101628*x^7 + 20920*x^6 - 12933*x^5 - 2085*x^4 + 755*x^3 + 90*x^2 - 15*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -15, 90, 755, -2085, -12933, 20920, 101628, -102465, -414755, 259620, 930681, -344712, -1168977, 238428, 825846, -91596, -339003, 19984, 83283, -2448, -12356, 156, 1080, -4, -51, 0, 1]);
 

\( x^{27} - 51 x^{25} - 4 x^{24} + 1080 x^{23} + 156 x^{22} - 12356 x^{21} - 2448 x^{20} + 83283 x^{19} + 19984 x^{18} - 339003 x^{17} - 91596 x^{16} + 825846 x^{15} + 238428 x^{14} - 1168977 x^{13} - 344712 x^{12} + 930681 x^{11} + 259620 x^{10} - 414755 x^{9} - 102465 x^{8} + 101628 x^{7} + 20920 x^{6} - 12933 x^{5} - 2085 x^{4} + 755 x^{3} + 90 x^{2} - 15 x - 1 \)

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

Invariants

Degree:  $27$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[27, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(735278035529026786878794063569162978753987707441\)\(\medspace = 3^{36}\cdot 19^{24}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $59.27$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 19$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $27$
This field is Galois and abelian over $\Q$.
Conductor:  \(171=3^{2}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{171}(64,·)$, $\chi_{171}(1,·)$, $\chi_{171}(130,·)$, $\chi_{171}(4,·)$, $\chi_{171}(7,·)$, $\chi_{171}(73,·)$, $\chi_{171}(139,·)$, $\chi_{171}(142,·)$, $\chi_{171}(16,·)$, $\chi_{171}(82,·)$, $\chi_{171}(85,·)$, $\chi_{171}(25,·)$, $\chi_{171}(28,·)$, $\chi_{171}(157,·)$, $\chi_{171}(163,·)$, $\chi_{171}(100,·)$, $\chi_{171}(169,·)$, $\chi_{171}(106,·)$, $\chi_{171}(43,·)$, $\chi_{171}(112,·)$, $\chi_{171}(49,·)$, $\chi_{171}(115,·)$, $\chi_{171}(118,·)$, $\chi_{171}(55,·)$, $\chi_{171}(121,·)$, $\chi_{171}(58,·)$, $\chi_{171}(61,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{37} a^{22} + \frac{7}{37} a^{21} - \frac{18}{37} a^{19} - \frac{4}{37} a^{18} - \frac{11}{37} a^{17} + \frac{1}{37} a^{16} + \frac{6}{37} a^{15} - \frac{1}{37} a^{14} - \frac{18}{37} a^{13} + \frac{1}{37} a^{11} - \frac{6}{37} a^{10} + \frac{9}{37} a^{9} - \frac{6}{37} a^{8} - \frac{9}{37} a^{7} - \frac{13}{37} a^{6} - \frac{10}{37} a^{5} - \frac{15}{37} a^{4} + \frac{8}{37} a^{3} + \frac{11}{37} a^{2} + \frac{11}{37} a + \frac{8}{37}$, $\frac{1}{37} a^{23} - \frac{12}{37} a^{21} - \frac{18}{37} a^{20} + \frac{11}{37} a^{19} + \frac{17}{37} a^{18} + \frac{4}{37} a^{17} - \frac{1}{37} a^{16} - \frac{6}{37} a^{15} - \frac{11}{37} a^{14} + \frac{15}{37} a^{13} + \frac{1}{37} a^{12} - \frac{13}{37} a^{11} + \frac{14}{37} a^{10} + \frac{5}{37} a^{9} - \frac{4}{37} a^{8} + \frac{13}{37} a^{7} + \frac{7}{37} a^{6} + \frac{18}{37} a^{5} + \frac{2}{37} a^{4} - \frac{8}{37} a^{3} + \frac{8}{37} a^{2} + \frac{5}{37} a + \frac{18}{37}$, $\frac{1}{259} a^{24} + \frac{1}{259} a^{23} - \frac{1}{259} a^{22} + \frac{10}{259} a^{21} + \frac{104}{259} a^{20} + \frac{18}{37} a^{19} + \frac{51}{259} a^{18} - \frac{81}{259} a^{17} - \frac{10}{37} a^{16} - \frac{62}{259} a^{15} + \frac{67}{259} a^{14} + \frac{3}{259} a^{13} + \frac{25}{259} a^{12} + \frac{86}{259} a^{11} - \frac{12}{37} a^{10} + \frac{9}{37} a^{9} + \frac{54}{259} a^{8} - \frac{5}{259} a^{7} + \frac{67}{259} a^{6} - \frac{53}{259} a^{5} + \frac{2}{37} a^{4} - \frac{97}{259} a^{3} - \frac{2}{37} a^{2} - \frac{41}{259} a + \frac{106}{259}$, $\frac{1}{167573} a^{25} + \frac{207}{167573} a^{24} - \frac{1664}{167573} a^{23} + \frac{141}{23939} a^{22} - \frac{24625}{167573} a^{21} + \frac{34213}{167573} a^{20} - \frac{75675}{167573} a^{19} - \frac{40066}{167573} a^{18} - \frac{71951}{167573} a^{17} + \frac{80515}{167573} a^{16} - \frac{10225}{23939} a^{15} + \frac{82909}{167573} a^{14} + \frac{57504}{167573} a^{13} + \frac{178}{647} a^{12} - \frac{48574}{167573} a^{11} + \frac{314}{647} a^{10} - \frac{81496}{167573} a^{9} - \frac{1453}{167573} a^{8} + \frac{12526}{167573} a^{7} + \frac{15331}{167573} a^{6} + \frac{81671}{167573} a^{5} - \frac{27243}{167573} a^{4} - \frac{6717}{167573} a^{3} - \frac{32059}{167573} a^{2} - \frac{12183}{167573} a - \frac{35753}{167573}$, $\frac{1}{22535003470568102738975368344886506403} a^{26} + \frac{13054259234559289784144812899643}{22535003470568102738975368344886506403} a^{25} - \frac{17937507015278378530930310980190101}{22535003470568102738975368344886506403} a^{24} + \frac{10311887617202206858569341358683003}{3219286210081157534139338334983786629} a^{23} + \frac{61237083266114227258150526553303285}{22535003470568102738975368344886506403} a^{22} + \frac{4949653980576477565314222355122851781}{22535003470568102738975368344886506403} a^{21} + \frac{10273305630368989160359027009381756363}{22535003470568102738975368344886506403} a^{20} + \frac{9205919698363090412057030368987904144}{22535003470568102738975368344886506403} a^{19} - \frac{6537077146895370159555042498736814178}{22535003470568102738975368344886506403} a^{18} + \frac{3998484399830831854030415746816005019}{22535003470568102738975368344886506403} a^{17} - \frac{6837612767192994241052033309831389463}{22535003470568102738975368344886506403} a^{16} + \frac{793095610656784969862086804832390409}{3219286210081157534139338334983786629} a^{15} - \frac{3565826732592495848822811171484654821}{22535003470568102738975368344886506403} a^{14} - \frac{1599666878031742865156627947071896865}{3219286210081157534139338334983786629} a^{13} + \frac{9653194163033856187240490483732394}{34829989908142353537829008261030149} a^{12} - \frac{8598841684481909585504240376488359114}{22535003470568102738975368344886506403} a^{11} - \frac{2873366968198295144958993155938251014}{22535003470568102738975368344886506403} a^{10} + \frac{1497150057525618447779254771726818963}{22535003470568102738975368344886506403} a^{9} - \frac{7618644660018565106267151869599299350}{22535003470568102738975368344886506403} a^{8} + \frac{7940705255687416446131418699247905182}{22535003470568102738975368344886506403} a^{7} - \frac{2858004253349434195268988977742357522}{22535003470568102738975368344886506403} a^{6} + \frac{5467886876102442685586029906292795329}{22535003470568102738975368344886506403} a^{5} - \frac{527504563821138192337928847090142043}{3219286210081157534139338334983786629} a^{4} - \frac{2060382369645091529607610590313499878}{22535003470568102738975368344886506403} a^{3} + \frac{1253859824879841205113139674900987073}{22535003470568102738975368344886506403} a^{2} + \frac{5140224931908462363741447822331475639}{22535003470568102738975368344886506403} a - \frac{2570516716407076236321134730778802312}{22535003470568102738975368344886506403}$

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

Class group and class number

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

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $26$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 3059088874781005.5 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{27}\cdot(2\pi)^{0}\cdot 3059088874781005.5 \cdot 1}{2\sqrt{735278035529026786878794063569162978753987707441}}\approx 0.239412149281721$ (assuming GRH)

Galois group

$C_3\times C_9$ (as 27T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
An abelian group of order 27
The 27 conjugacy class representatives for $C_3\times C_9$
Character table for $C_3\times C_9$ is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.3.361.1, 3.3.29241.1, 3.3.29241.2, 9.9.25002110044521.1, 9.9.9025761726072081.1, 9.9.9025761726072081.2, \(\Q(\zeta_{19})^+\)

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

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{27}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
$19$19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$