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Properties

Label 27.27.2748591706...4441.1
Degree $27$
Signature $[27, 0]$
Discriminant $3^{36}\cdot 7^{18}\cdot 13^{18}$
Root discriminant $87.54$
Ramified primes $3, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3^3$ (as 27T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7064, 16716, -528300, -134797, 7157724, 2104236, -37059379, -15697422, 88142967, 44676536, -103930188, -50585529, 71864499, 28970679, -31587348, -9244050, 9058716, 1662003, -1689790, -152250, 200187, 3278, -14325, 564, 558, -45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 9*x^26 - 45*x^25 + 558*x^24 + 564*x^23 - 14325*x^22 + 3278*x^21 + 200187*x^20 - 152250*x^19 - 1689790*x^18 + 1662003*x^17 + 9058716*x^16 - 9244050*x^15 - 31587348*x^14 + 28970679*x^13 + 71864499*x^12 - 50585529*x^11 - 103930188*x^10 + 44676536*x^9 + 88142967*x^8 - 15697422*x^7 - 37059379*x^6 + 2104236*x^5 + 7157724*x^4 - 134797*x^3 - 528300*x^2 + 16716*x + 7064)
 
gp: K = bnfinit(x^27 - 9*x^26 - 45*x^25 + 558*x^24 + 564*x^23 - 14325*x^22 + 3278*x^21 + 200187*x^20 - 152250*x^19 - 1689790*x^18 + 1662003*x^17 + 9058716*x^16 - 9244050*x^15 - 31587348*x^14 + 28970679*x^13 + 71864499*x^12 - 50585529*x^11 - 103930188*x^10 + 44676536*x^9 + 88142967*x^8 - 15697422*x^7 - 37059379*x^6 + 2104236*x^5 + 7157724*x^4 - 134797*x^3 - 528300*x^2 + 16716*x + 7064, 1)
 

Normalized defining polynomial

\( x^{27} - 9 x^{26} - 45 x^{25} + 558 x^{24} + 564 x^{23} - 14325 x^{22} + 3278 x^{21} + 200187 x^{20} - 152250 x^{19} - 1689790 x^{18} + 1662003 x^{17} + 9058716 x^{16} - 9244050 x^{15} - 31587348 x^{14} + 28970679 x^{13} + 71864499 x^{12} - 50585529 x^{11} - 103930188 x^{10} + 44676536 x^{9} + 88142967 x^{8} - 15697422 x^{7} - 37059379 x^{6} + 2104236 x^{5} + 7157724 x^{4} - 134797 x^{3} - 528300 x^{2} + 16716 x + 7064 \)

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

Invariants

Degree:  $27$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[27, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(27485917061841905258715957728464424188465393850574441=3^{36}\cdot 7^{18}\cdot 13^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $87.54$
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:  $3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(819=3^{2}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{819}(256,·)$, $\chi_{819}(1,·)$, $\chi_{819}(646,·)$, $\chi_{819}(781,·)$, $\chi_{819}(718,·)$, $\chi_{819}(79,·)$, $\chi_{819}(16,·)$, $\chi_{819}(529,·)$, $\chi_{819}(274,·)$, $\chi_{819}(211,·)$, $\chi_{819}(22,·)$, $\chi_{819}(100,·)$, $\chi_{819}(352,·)$, $\chi_{819}(289,·)$, $\chi_{819}(802,·)$, $\chi_{819}(547,·)$, $\chi_{819}(484,·)$, $\chi_{819}(295,·)$, $\chi_{819}(235,·)$, $\chi_{819}(172,·)$, $\chi_{819}(625,·)$, $\chi_{819}(562,·)$, $\chi_{819}(757,·)$, $\chi_{819}(568,·)$, $\chi_{819}(508,·)$, $\chi_{819}(445,·)$, $\chi_{819}(373,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{3}$, $\frac{1}{24} a^{18} - \frac{1}{8} a^{16} + \frac{1}{12} a^{15} - \frac{1}{8} a^{14} + \frac{1}{12} a^{12} + \frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{5}{24} a^{9} - \frac{1}{4} a^{8} - \frac{3}{8} a^{7} - \frac{1}{12} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} + \frac{11}{24} a^{3} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{24} a^{19} - \frac{1}{8} a^{17} + \frac{1}{12} a^{16} - \frac{1}{8} a^{15} + \frac{1}{12} a^{13} + \frac{1}{8} a^{12} - \frac{1}{8} a^{11} + \frac{5}{24} a^{10} - \frac{1}{4} a^{9} + \frac{1}{8} a^{8} - \frac{1}{12} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} + \frac{11}{24} a^{4} - \frac{1}{2} a^{2} + \frac{1}{6} a$, $\frac{1}{24} a^{20} + \frac{1}{12} a^{17} - \frac{1}{24} a^{14} - \frac{1}{8} a^{13} + \frac{1}{8} a^{12} - \frac{1}{6} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{12} a^{8} + \frac{1}{8} a^{7} + \frac{1}{4} a^{6} - \frac{1}{24} a^{5} - \frac{1}{8} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{24} a^{21} + \frac{1}{24} a^{15} - \frac{1}{8} a^{14} - \frac{1}{8} a^{13} + \frac{1}{6} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{8} + \frac{1}{4} a^{7} - \frac{1}{8} a^{6} - \frac{1}{2} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{3}$, $\frac{1}{24} a^{22} + \frac{1}{24} a^{16} - \frac{1}{8} a^{15} - \frac{1}{8} a^{14} - \frac{1}{12} a^{13} + \frac{1}{8} a^{12} - \frac{1}{4} a^{11} + \frac{1}{8} a^{9} - \frac{1}{4} a^{8} + \frac{3}{8} a^{7} + \frac{1}{4} a^{6} - \frac{1}{8} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{3} a$, $\frac{1}{24} a^{23} + \frac{1}{24} a^{17} - \frac{1}{8} a^{16} - \frac{1}{8} a^{15} - \frac{1}{12} a^{14} - \frac{1}{8} a^{13} - \frac{1}{4} a^{11} + \frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{3}{8} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{5}{12} a^{2}$, $\frac{1}{6096} a^{24} + \frac{13}{2032} a^{23} - \frac{61}{3048} a^{22} + \frac{2}{127} a^{21} - \frac{79}{6096} a^{20} + \frac{1}{6096} a^{19} - \frac{121}{6096} a^{18} + \frac{691}{6096} a^{17} + \frac{45}{508} a^{16} - \frac{15}{254} a^{15} - \frac{25}{3048} a^{14} - \frac{89}{1016} a^{13} - \frac{167}{3048} a^{12} + \frac{34}{381} a^{11} - \frac{595}{6096} a^{10} + \frac{521}{6096} a^{9} - \frac{239}{3048} a^{8} + \frac{17}{3048} a^{7} - \frac{5}{48} a^{6} - \frac{1415}{6096} a^{5} + \frac{2021}{6096} a^{4} - \frac{87}{2032} a^{3} + \frac{71}{1524} a^{2} + \frac{761}{1524} a + \frac{251}{762}$, $\frac{1}{1871472} a^{25} + \frac{13}{935736} a^{24} + \frac{4451}{1871472} a^{23} + \frac{611}{467868} a^{22} + \frac{14929}{1871472} a^{21} - \frac{4253}{233934} a^{20} + \frac{2473}{935736} a^{19} + \frac{19039}{935736} a^{18} + \frac{36979}{623824} a^{17} + \frac{13223}{116967} a^{16} - \frac{13055}{155956} a^{15} + \frac{3061}{38989} a^{14} + \frac{57913}{467868} a^{13} - \frac{3965}{116967} a^{12} + \frac{273511}{1871472} a^{11} - \frac{98615}{935736} a^{10} + \frac{44319}{623824} a^{9} + \frac{2486}{116967} a^{8} + \frac{150307}{1871472} a^{7} + \frac{71055}{155956} a^{6} + \frac{150797}{935736} a^{5} - \frac{36765}{311912} a^{4} - \frac{621925}{1871472} a^{3} - \frac{35849}{467868} a^{2} - \frac{36235}{155956} a + \frac{54395}{233934}$, $\frac{1}{214044343958580802294436213597101671102430918838318343426384} a^{26} + \frac{16397944620223731083155739058936248223205909293449145}{71348114652860267431478737865700557034143639612772781142128} a^{25} + \frac{1990882126419155695398429659181875452788253676353263165}{71348114652860267431478737865700557034143639612772781142128} a^{24} - \frac{491953650021808417247791412165946562701036575550760950415}{71348114652860267431478737865700557034143639612772781142128} a^{23} + \frac{2441367805138205965948826784922947195072433007081458789061}{214044343958580802294436213597101671102430918838318343426384} a^{22} + \frac{4449117363724745002353833191375165047172959627550640846359}{214044343958580802294436213597101671102430918838318343426384} a^{21} - \frac{247526684553597566120069826206389158016453406729927579005}{17837028663215066857869684466425139258535909903193195285532} a^{20} - \frac{1632765869280575012237754462343045227280199732203855728661}{107022171979290401147218106798550835551215459419159171713192} a^{19} - \frac{1028547438293340163727586933312283786834083147822741791553}{71348114652860267431478737865700557034143639612772781142128} a^{18} - \frac{10431000162317193849655760102442267641608529015983580157967}{214044343958580802294436213597101671102430918838318343426384} a^{17} + \frac{1253059853206262818070736418959785031702716528240370709631}{107022171979290401147218106798550835551215459419159171713192} a^{16} + \frac{481394163101998734545936227248592793260201597078189886081}{26755542994822600286804526699637708887803864854789792928298} a^{15} + \frac{3834618596977391896715166722267500571670446264231725757927}{107022171979290401147218106798550835551215459419159171713192} a^{14} + \frac{260459660591981749971259162812122579923444295739381590374}{13377771497411300143402263349818854443901932427394896464149} a^{13} - \frac{21786974872777466300733249588987460851567729988592757395393}{214044343958580802294436213597101671102430918838318343426384} a^{12} + \frac{26667534440963774373901285688802753150439771578607996577731}{214044343958580802294436213597101671102430918838318343426384} a^{11} + \frac{49499194253781204339537556559369673242725013206483635633719}{214044343958580802294436213597101671102430918838318343426384} a^{10} + \frac{16456399431470734569951347629490815107432607334265872599893}{71348114652860267431478737865700557034143639612772781142128} a^{9} - \frac{32396090369104066356226039296833656190442110103391868103769}{214044343958580802294436213597101671102430918838318343426384} a^{8} - \frac{18129725364734494273173249575339760198145960672421869306529}{214044343958580802294436213597101671102430918838318343426384} a^{7} - \frac{72127229606663800704538770851314880993400254882166217459}{17837028663215066857869684466425139258535909903193195285532} a^{6} + \frac{1889481174763927856430387978332257366269476742096082564873}{107022171979290401147218106798550835551215459419159171713192} a^{5} + \frac{32933469565356758891287543722563482919154548336033563699577}{214044343958580802294436213597101671102430918838318343426384} a^{4} + \frac{6684182086145207293443234600673780059821104235152560075511}{71348114652860267431478737865700557034143639612772781142128} a^{3} + \frac{8498190766351899502982869346212052128698640739826721421717}{53511085989645200573609053399275417775607729709579585856596} a^{2} + \frac{3613624388009575298762669518710062647202674088322469552769}{17837028663215066857869684466425139258535909903193195285532} a - \frac{12392495731036406311982807150170938132382460492519484415285}{26755542994822600286804526699637708887803864854789792928298}$

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:  $26$
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:  \( 1149187264584812300 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^3$ (as 27T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 27
The 27 conjugacy class representatives for $C_3^3$
Character table for $C_3^3$ is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.3.13689.2, 3.3.13689.1, 3.3.169.1, 3.3.670761.2, 3.3.670761.4, 3.3.8281.2, \(\Q(\zeta_{7})^+\), 3.3.3969.2, 3.3.3969.1, 3.3.670761.1, 3.3.8281.1, 3.3.670761.3, 9.9.2565164201769.1, 9.9.301789003173921081.10, 9.9.62523502209.1, 9.9.301789003173921081.12, 9.9.301789003173921081.1, 9.9.301789003173921081.5, 9.9.301789003173921081.6, 9.9.301789003173921081.3, 9.9.301789003173921081.9, 9.9.301789003173921081.4, 9.9.301789003173921081.7, 9.9.301789003173921081.8, 9.9.567869252041.1

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{9}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{9}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{9}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{9}$

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$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$13$13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$