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Properties

Label 27.27.3063651216...3609.1
Degree $27$
Signature $[27, 0]$
Discriminant $3^{94}\cdot 37^{24}$
Root discriminant $1135.13$
Ramified primes $3, 37$
Class number Not computed
Class group Not computed
Galois group $C_{27}$ (as 27T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2080943940400297569469, 9541558188138634418511, -12388857051128754195612, 3939372175488139375839, 2232226495698874629840, -1276110277553555013213, -154848144296228240088, 148209351279180829002, 5380823546973182088, -9477154119242768340, -105031090256233284, 377261368227838926, 1204287685165656, -9934810805722716, -8047660503780, 179005600103112, 29000578392, -2241127346283, -43544412, 19481903595, 0, -115336881, 0, 443556, 0, -999, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 999*x^25 + 443556*x^23 - 115336881*x^21 + 19481903595*x^19 - 43544412*x^18 - 2241127346283*x^17 + 29000578392*x^16 + 179005600103112*x^15 - 8047660503780*x^14 - 9934810805722716*x^13 + 1204287685165656*x^12 + 377261368227838926*x^11 - 105031090256233284*x^10 - 9477154119242768340*x^9 + 5380823546973182088*x^8 + 148209351279180829002*x^7 - 154848144296228240088*x^6 - 1276110277553555013213*x^5 + 2232226495698874629840*x^4 + 3939372175488139375839*x^3 - 12388857051128754195612*x^2 + 9541558188138634418511*x - 2080943940400297569469)
 
gp: K = bnfinit(x^27 - 999*x^25 + 443556*x^23 - 115336881*x^21 + 19481903595*x^19 - 43544412*x^18 - 2241127346283*x^17 + 29000578392*x^16 + 179005600103112*x^15 - 8047660503780*x^14 - 9934810805722716*x^13 + 1204287685165656*x^12 + 377261368227838926*x^11 - 105031090256233284*x^10 - 9477154119242768340*x^9 + 5380823546973182088*x^8 + 148209351279180829002*x^7 - 154848144296228240088*x^6 - 1276110277553555013213*x^5 + 2232226495698874629840*x^4 + 3939372175488139375839*x^3 - 12388857051128754195612*x^2 + 9541558188138634418511*x - 2080943940400297569469, 1)
 

Normalized defining polynomial

\( x^{27} - 999 x^{25} + 443556 x^{23} - 115336881 x^{21} + 19481903595 x^{19} - 43544412 x^{18} - 2241127346283 x^{17} + 29000578392 x^{16} + 179005600103112 x^{15} - 8047660503780 x^{14} - 9934810805722716 x^{13} + 1204287685165656 x^{12} + 377261368227838926 x^{11} - 105031090256233284 x^{10} - 9477154119242768340 x^{9} + 5380823546973182088 x^{8} + 148209351279180829002 x^{7} - 154848144296228240088 x^{6} - 1276110277553555013213 x^{5} + 2232226495698874629840 x^{4} + 3939372175488139375839 x^{3} - 12388857051128754195612 x^{2} + 9541558188138634418511 x - 2080943940400297569469 \)

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:  \(30636512167696967158640831949436414926745966008852301931537630975865004722034443609=3^{94}\cdot 37^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1135.13$
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:  $3, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2997=3^{4}\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{2997}(1,·)$, $\chi_{2997}(1099,·)$, $\chi_{2997}(1348,·)$, $\chi_{2997}(1921,·)$, $\chi_{2997}(10,·)$, $\chi_{2997}(1291,·)$, $\chi_{2997}(1228,·)$, $\chi_{2997}(1933,·)$, $\chi_{2997}(1999,·)$, $\chi_{2997}(1492,·)$, $\chi_{2997}(2008,·)$, $\chi_{2997}(100,·)$, $\chi_{2997}(922,·)$, $\chi_{2997}(349,·)$, $\chi_{2997}(292,·)$, $\chi_{2997}(229,·)$, $\chi_{2997}(934,·)$, $\chi_{2997}(1009,·)$, $\chi_{2997}(2920,·)$, $\chi_{2997}(2290,·)$, $\chi_{2997}(2347,·)$, $\chi_{2997}(493,·)$, $\chi_{2997}(1000,·)$, $\chi_{2997}(2098,·)$, $\chi_{2997}(2227,·)$, $\chi_{2997}(2932,·)$, $\chi_{2997}(2491,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{37} a^{9}$, $\frac{1}{37} a^{10}$, $\frac{1}{37} a^{11}$, $\frac{1}{37} a^{12}$, $\frac{1}{37} a^{13}$, $\frac{1}{1369} a^{14} - \frac{17}{37} a^{5}$, $\frac{1}{1369} a^{15} - \frac{17}{37} a^{6}$, $\frac{1}{1369} a^{16} - \frac{17}{37} a^{7}$, $\frac{1}{1369} a^{17} - \frac{17}{37} a^{8}$, $\frac{1}{3239531404691195231} a^{18} - \frac{18}{87554902829491763} a^{16} + \frac{135}{2366348725121399} a^{14} - \frac{546}{63955370949227} a^{12} + \frac{47619}{63955370949227} a^{10} - \frac{1090175382754021}{87554902829491763} a^{9} - \frac{2439558}{63955370949227} a^{8} + \frac{346183544300593}{2366348725121399} a^{7} + \frac{70205058}{63955370949227} a^{6} - \frac{15264697714147}{63955370949227} a^{5} - \frac{1012046940}{63955370949227} a^{4} - \frac{4898873360757}{63955370949227} a^{3} + \frac{5616860517}{63955370949227} a^{2} - \frac{15973413739747}{63955370949227} a - \frac{20477765767421}{63955370949227}$, $\frac{1}{3239531404691195231} a^{19} - \frac{18}{87554902829491763} a^{17} + \frac{135}{2366348725121399} a^{15} - \frac{546}{63955370949227} a^{13} + \frac{47619}{63955370949227} a^{11} - \frac{1090175382754021}{87554902829491763} a^{10} - \frac{2439558}{63955370949227} a^{9} + \frac{346183544300593}{2366348725121399} a^{8} + \frac{70205058}{63955370949227} a^{7} - \frac{15264697714147}{63955370949227} a^{6} - \frac{1012046940}{63955370949227} a^{5} - \frac{4898873360757}{63955370949227} a^{4} + \frac{5616860517}{63955370949227} a^{3} - \frac{15973413739747}{63955370949227} a^{2} - \frac{20477765767421}{63955370949227} a$, $\frac{1}{3239531404691195231} a^{20} - \frac{189}{2366348725121399} a^{16} + \frac{1884}{63955370949227} a^{14} - \frac{316017}{63955370949227} a^{12} - \frac{1090175382754021}{87554902829491763} a^{11} + \frac{29274696}{63955370949227} a^{10} - \frac{26406689554458}{2366348725121399} a^{9} - \frac{1554540570}{63955370949227} a^{8} + \frac{12368117621508}{63955370949227} a^{7} + \frac{45744521688}{63955370949227} a^{6} - \frac{2283570055566}{63955370949227} a^{5} - \frac{668406401523}{63955370949227} a^{4} - \frac{16899153593332}{63955370949227} a^{3} - \frac{16736936663099}{63955370949227} a^{2} - \frac{21701973099820}{63955370949227} a - \frac{15697988917035}{63955370949227}$, $\frac{1}{119862661973574223547} a^{21} - \frac{189}{87554902829491763} a^{17} + \frac{1884}{2366348725121399} a^{15} - \frac{8541}{63955370949227} a^{13} + \frac{27306009318702767}{3239531404691195231} a^{12} + \frac{791208}{63955370949227} a^{11} - \frac{346183544300593}{87554902829491763} a^{10} - \frac{42014610}{63955370949227} a^{9} + \frac{460055714266097}{2366348725121399} a^{8} + \frac{1236338424}{63955370949227} a^{7} + \frac{5123852507895}{63955370949227} a^{6} - \frac{18065037879}{63955370949227} a^{5} + \frac{22014072128287}{63955370949227} a^{4} - \frac{784201388053823}{2366348725121399} a^{3} - \frac{31699963518538}{63955370949227} a^{2} + \frac{15132441881784}{63955370949227} a$, $\frac{1}{119862661973574223547} a^{22} - \frac{1518}{2366348725121399} a^{16} + \frac{16974}{63955370949227} a^{14} + \frac{27306009318702767}{3239531404691195231} a^{13} - \frac{3026970}{63955370949227} a^{12} - \frac{346183544300593}{87554902829491763} a^{11} + \frac{290985057}{63955370949227} a^{10} - \frac{30529395428294}{2366348725121399} a^{9} - \frac{15823490670}{63955370949227} a^{8} + \frac{7469244260751}{63955370949227} a^{7} + \frac{472878932715}{63955370949227} a^{6} + \frac{17497071358179}{63955370949227} a^{5} - \frac{1046059425356363}{2366348725121399} a^{4} - \frac{9442546506567}{63955370949227} a^{3} - \frac{9544223472062}{63955370949227} a^{2} + \frac{27950766248798}{63955370949227} a - \frac{4940456255800}{63955370949227}$, $\frac{1}{34345427669002829238702127222031} a^{23} + \frac{1552458225}{928254801864941330775733168163} a^{22} - \frac{23}{928254801864941330775733168163} a^{21} - \frac{3176801234}{25087967617971387318263058599} a^{20} + \frac{230}{25087967617971387318263058599} a^{19} + \frac{1463563975}{25087967617971387318263058599} a^{18} - \frac{1311}{678053178864091549142244827} a^{17} - \frac{65006341787538}{678053178864091549142244827} a^{16} + \frac{4692}{18325761590921393220060671} a^{15} - \frac{264893115917093498700722514}{928254801864941330775733168163} a^{14} - \frac{266818186019111243278717947}{25087967617971387318263058599} a^{13} + \frac{318238075816380723262472321}{25087967617971387318263058599} a^{12} + \frac{7948332975076953170246058}{678053178864091549142244827} a^{11} - \frac{199723551265758730945764}{678053178864091549142244827} a^{10} + \frac{5055726172069455461822238}{678053178864091549142244827} a^{9} + \frac{8875205574680922070782305}{18325761590921393220060671} a^{8} - \frac{3792671826480720449430234}{18325761590921393220060671} a^{7} - \frac{43444408407429525697475}{495290853808686303244883} a^{6} - \frac{7086408822364248731863718}{678053178864091549142244827} a^{5} + \frac{122662560121237956041233}{18325761590921393220060671} a^{4} - \frac{8421124810870374637722699}{18325761590921393220060671} a^{3} + \frac{247339832852159499490033}{495290853808686303244883} a^{2} - \frac{62178021050901766043612}{495290853808686303244883} a + \frac{168594775190953721152843}{495290853808686303244883}$, $\frac{1}{34345427669002829238702127222031} a^{24} - \frac{24}{928254801864941330775733168163} a^{22} + \frac{3230714822}{928254801864941330775733168163} a^{21} + \frac{252}{25087967617971387318263058599} a^{20} + \frac{1853868099}{25087967617971387318263058599} a^{19} - \frac{1520}{678053178864091549142244827} a^{18} - \frac{643974727140}{678053178864091549142244827} a^{17} + \frac{5814}{18325761590921393220060671} a^{16} - \frac{264893153775902480429225484}{928254801864941330775733168163} a^{15} - \frac{14688}{495290853808686303244883} a^{14} - \frac{94921763180204500012921035}{25087967617971387318263058599} a^{13} + \frac{174365852944413544674340919}{25087967617971387318263058599} a^{12} + \frac{1431964689109920951646234}{678053178864091549142244827} a^{11} - \frac{5934370089142140527759983}{678053178864091549142244827} a^{10} + \frac{219756688087156155372973}{18325761590921393220060671} a^{9} + \frac{2117444764781865873017351}{18325761590921393220060671} a^{8} + \frac{225737515397727586713913}{495290853808686303244883} a^{7} - \frac{88309130295843903180272413}{678053178864091549142244827} a^{6} - \frac{240275151051640066299423}{495290853808686303244883} a^{5} + \frac{2335482340206439866058443}{18325761590921393220060671} a^{4} - \frac{2796078617765878122972303}{18325761590921393220060671} a^{3} - \frac{171219507631771695861743}{495290853808686303244883} a^{2} + \frac{220925169326066112741796}{495290853808686303244883} a + \frac{56517831680010995472291}{495290853808686303244883}$, $\frac{1}{1270780823753104681831978707215147} a^{25} + \frac{2428540}{25087967617971387318263058599} a^{22} - \frac{300}{928254801864941330775733168163} a^{21} - \frac{424373335}{25087967617971387318263058599} a^{20} + \frac{4000}{25087967617971387318263058599} a^{19} + \frac{2952082651}{25087967617971387318263058599} a^{18} - \frac{25650}{678053178864091549142244827} a^{17} - \frac{8401531300573995745348295076}{34345427669002829238702127222031} a^{16} + \frac{97920}{18325761590921393220060671} a^{15} + \frac{328174329964637186928068636}{928254801864941330775733168163} a^{14} - \frac{7275730911195183624936789}{678053178864091549142244827} a^{13} + \frac{70610729815733533268105763}{25087967617971387318263058599} a^{12} + \frac{7474342263606736383849710}{678053178864091549142244827} a^{11} - \frac{4090082644527537056166141}{678053178864091549142244827} a^{10} - \frac{6867151731771216496279520}{678053178864091549142244827} a^{9} + \frac{273003386443331853142637}{18325761590921393220060671} a^{8} - \frac{379509873944398224376466041}{25087967617971387318263058599} a^{7} + \frac{61175538496841432384817}{495290853808686303244883} a^{6} + \frac{321980913897592063765653362}{678053178864091549142244827} a^{5} + \frac{182690932132056568546968}{495290853808686303244883} a^{4} - \frac{75925136313395215811896}{495290853808686303244883} a^{3} - \frac{80532905792388523183725}{495290853808686303244883} a^{2} + \frac{146732134134753923881177}{495290853808686303244883} a + \frac{150690577865126248921511}{495290853808686303244883}$, $\frac{1}{1270780823753104681831978707215147} a^{26} - \frac{325}{928254801864941330775733168163} a^{22} - \frac{1188933847}{928254801864941330775733168163} a^{21} + \frac{4550}{25087967617971387318263058599} a^{20} - \frac{2778954107}{25087967617971387318263058599} a^{19} - \frac{30875}{678053178864091549142244827} a^{18} - \frac{8401531300452070155124657020}{34345427669002829238702127222031} a^{17} + \frac{125970}{18325761590921393220060671} a^{16} + \frac{328174328451915994026845522}{928254801864941330775733168163} a^{15} - \frac{331500}{495290853808686303244883} a^{14} - \frac{87592198745984494067850182}{25087967617971387318263058599} a^{13} + \frac{173221446331292324918001168}{25087967617971387318263058599} a^{12} + \frac{62769567076537927254210}{678053178864091549142244827} a^{11} - \frac{1293152986031918051907141}{678053178864091549142244827} a^{10} - \frac{184404344476281839791317}{18325761590921393220060671} a^{9} - \frac{2371982084697741945500894770}{25087967617971387318263058599} a^{8} - \frac{128753723278390452810082}{495290853808686303244883} a^{7} + \frac{162998952926049797450447580}{678053178864091549142244827} a^{6} - \frac{13430845508286209813497}{495290853808686303244883} a^{5} + \frac{7007900442145502663289397}{18325761590921393220060671} a^{4} + \frac{6857512832350622271574278}{18325761590921393220060671} a^{3} - \frac{102934641490210891192056}{495290853808686303244883} a^{2} + \frac{195803949144130386660773}{495290853808686303244883} a - \frac{72926769425784022927374}{495290853808686303244883}$

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:  $26$
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:  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

$C_{27}$ (as 27T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 27
The 27 conjugacy class representatives for $C_{27}$
Character table for $C_{27}$ is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.9.80515213381214514081.2

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 $27$ R $27$ $27$ $27$ $27$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{3}$ $27$ $27$ $27$ R $27$ $27$ $27$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ $27$

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
3Data not computed
$37$37.9.8.8$x^{9} + 4736$$9$$1$$8$$C_9$$[\ ]_{9}$
37.9.8.8$x^{9} + 4736$$9$$1$$8$$C_9$$[\ ]_{9}$
37.9.8.8$x^{9} + 4736$$9$$1$$8$$C_9$$[\ ]_{9}$