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Properties

Label 27.27.1785789002...6481.1
Degree $27$
Signature $[27, 0]$
Discriminant $3^{94}\cdot 43^{18}$
Root discriminant $562.43$
Ramified primes $3, 43$
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![-22298685735659897360231, -46393131355621851201561, 0, 32726937622958050072419, 0, -6849824153642382573297, 0, 667535465692723883112, 0, -36654079188553959990, 0, 1255382839016013006, 0, -28446361289569764, 0, 441028857202632, 0, -4751165869317, 0, 35538436395, 0, -181037439, 0, 599076, 0, -1161, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 1161*x^25 + 599076*x^23 - 181037439*x^21 + 35538436395*x^19 - 4751165869317*x^17 + 441028857202632*x^15 - 28446361289569764*x^13 + 1255382839016013006*x^11 - 36654079188553959990*x^9 + 667535465692723883112*x^7 - 6849824153642382573297*x^5 + 32726937622958050072419*x^3 - 46393131355621851201561*x - 22298685735659897360231)
 
gp: K = bnfinit(x^27 - 1161*x^25 + 599076*x^23 - 181037439*x^21 + 35538436395*x^19 - 4751165869317*x^17 + 441028857202632*x^15 - 28446361289569764*x^13 + 1255382839016013006*x^11 - 36654079188553959990*x^9 + 667535465692723883112*x^7 - 6849824153642382573297*x^5 + 32726937622958050072419*x^3 - 46393131355621851201561*x - 22298685735659897360231, 1)
 

Normalized defining polynomial

\( x^{27} - 1161 x^{25} + 599076 x^{23} - 181037439 x^{21} + 35538436395 x^{19} - 4751165869317 x^{17} + 441028857202632 x^{15} - 28446361289569764 x^{13} + 1255382839016013006 x^{11} - 36654079188553959990 x^{9} + 667535465692723883112 x^{7} - 6849824153642382573297 x^{5} + 32726937622958050072419 x^{3} - 46393131355621851201561 x - 22298685735659897360231 \)

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:  \(178578900240970840458250689871157457776932189215069373149025720786258346481=3^{94}\cdot 43^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $562.43$
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:  $3, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3483=3^{4}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{3483}(1,·)$, $\chi_{3483}(388,·)$, $\chi_{3483}(775,·)$, $\chi_{3483}(1162,·)$, $\chi_{3483}(2500,·)$, $\chi_{3483}(1549,·)$, $\chi_{3483}(1936,·)$, $\chi_{3483}(337,·)$, $\chi_{3483}(2323,·)$, $\chi_{3483}(724,·)$, $\chi_{3483}(2710,·)$, $\chi_{3483}(1111,·)$, $\chi_{3483}(3097,·)$, $\chi_{3483}(1498,·)$, $\chi_{3483}(1885,·)$, $\chi_{3483}(2272,·)$, $\chi_{3483}(2659,·)$, $\chi_{3483}(3046,·)$, $\chi_{3483}(2113,·)$, $\chi_{3483}(3433,·)$, $\chi_{3483}(2887,·)$, $\chi_{3483}(178,·)$, $\chi_{3483}(565,·)$, $\chi_{3483}(952,·)$, $\chi_{3483}(1339,·)$, $\chi_{3483}(3274,·)$, $\chi_{3483}(1726,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{43} a^{3}$, $\frac{1}{43} a^{4}$, $\frac{1}{43} a^{5}$, $\frac{1}{1849} a^{6}$, $\frac{1}{1849} a^{7}$, $\frac{1}{1849} a^{8}$, $\frac{1}{79507} a^{9}$, $\frac{1}{79507} a^{10}$, $\frac{1}{79507} a^{11}$, $\frac{1}{3418801} a^{12}$, $\frac{1}{3418801} a^{13}$, $\frac{1}{12778517454919} a^{14} - \frac{1787459}{12778517454919} a^{13} - \frac{14}{297174824533} a^{12} + \frac{810653}{297174824533} a^{11} + \frac{77}{6911042431} a^{10} + \frac{1382398}{297174824533} a^{9} - \frac{210}{160721917} a^{8} - \frac{327580}{6911042431} a^{7} + \frac{294}{3737719} a^{6} + \frac{1628083}{160721917} a^{5} - \frac{8428}{3737719} a^{4} + \frac{504546}{160721917} a^{3} + \frac{90601}{3737719} a^{2} - \frac{72078}{3737719} a - \frac{159014}{3737719}$, $\frac{1}{549476250561517} a^{15} - \frac{15}{12778517454919} a^{13} - \frac{1787459}{12778517454919} a^{12} + \frac{90}{297174824533} a^{11} - \frac{976806}{297174824533} a^{10} - \frac{275}{6911042431} a^{9} + \frac{657908}{6911042431} a^{8} + \frac{450}{160721917} a^{7} + \frac{435687}{6911042431} a^{6} - \frac{378}{3737719} a^{5} - \frac{642064}{160721917} a^{4} + \frac{6020}{3737719} a^{3} + \frac{1074472}{3737719} a^{2} - \frac{27735}{3737719} a - \frac{1736190}{3737719}$, $\frac{1}{549476250561517} a^{16} + \frac{1302408}{12778517454919} a^{13} - \frac{120}{297174824533} a^{12} - \frac{30168}{297174824533} a^{11} + \frac{880}{6911042431} a^{10} + \frac{435667}{297174824533} a^{9} - \frac{2700}{160721917} a^{8} - \frac{740294}{6911042431} a^{7} - \frac{20270}{6911042431} a^{6} + \frac{1352867}{160721917} a^{5} - \frac{1439481}{160721917} a^{4} + \frac{1442420}{160721917} a^{3} + \frac{1331280}{3737719} a^{2} + \frac{920359}{3737719} a + \frac{1352509}{3737719}$, $\frac{1}{549476250561517} a^{17} - \frac{136}{297174824533} a^{13} + \frac{1569121}{12778517454919} a^{12} + \frac{1088}{6911042431} a^{11} + \frac{1490505}{297174824533} a^{10} + \frac{560178}{297174824533} a^{9} + \frac{1139972}{6911042431} a^{8} + \frac{857115}{6911042431} a^{7} + \frac{299509}{6911042431} a^{6} + \frac{651669}{160721917} a^{5} + \frac{155932}{160721917} a^{4} + \frac{1072158}{160721917} a^{3} + \frac{1241981}{3737719} a^{2} - \frac{234071}{3737719} a + \frac{1571360}{3737719}$, $\frac{1}{23627478774145231} a^{18} - \frac{627740}{12778517454919} a^{13} - \frac{816}{297174824533} a^{12} - \frac{1587328}{297174824533} a^{11} + \frac{6732}{6911042431} a^{10} - \frac{1477569}{297174824533} a^{9} - \frac{22032}{160721917} a^{8} + \frac{1264874}{6911042431} a^{7} - \frac{172295}{6911042431} a^{6} + \frac{17006}{1474513} a^{5} - \frac{410892}{160721917} a^{4} - \frac{1156424}{160721917} a^{3} + \frac{668517}{3737719} a^{2} - \frac{204458}{3737719} a + \frac{800410}{3737719}$, $\frac{1}{23627478774145231} a^{19} - \frac{969}{297174824533} a^{13} - \frac{1366023}{12778517454919} a^{12} + \frac{8721}{6911042431} a^{11} - \frac{1202193}{297174824533} a^{10} + \frac{678031}{297174824533} a^{9} - \frac{845322}{6911042431} a^{8} - \frac{892991}{6911042431} a^{7} - \frac{1823799}{6911042431} a^{6} + \frac{429920}{160721917} a^{5} - \frac{376449}{160721917} a^{4} - \frac{544384}{160721917} a^{3} + \frac{534978}{3737719} a^{2} - \frac{354815}{3737719} a + \frac{75254}{3737719}$, $\frac{1}{23627478774145231} a^{20} - \frac{1631382}{12778517454919} a^{13} - \frac{1482967}{12778517454919} a^{12} - \frac{1490245}{297174824533} a^{11} + \frac{341865}{297174824533} a^{10} - \frac{581699}{297174824533} a^{9} + \frac{363618}{6911042431} a^{8} - \frac{949871}{6911042431} a^{7} - \frac{347973}{6911042431} a^{6} + \frac{1095781}{160721917} a^{5} - \frac{447092}{160721917} a^{4} - \frac{1173453}{160721917} a^{3} - \frac{379138}{3737719} a^{2} - \frac{1810415}{3737719} a + \frac{1339449}{3737719}$, $\frac{1}{1015981587288244933} a^{21} - \frac{5985}{297174824533} a^{13} + \frac{1834440}{12778517454919} a^{12} - \frac{1267111}{297174824533} a^{11} - \frac{20258}{6911042431} a^{10} + \frac{1648321}{297174824533} a^{9} + \frac{388601}{6911042431} a^{8} + \frac{72643}{6911042431} a^{7} + \frac{293583}{6911042431} a^{6} + \frac{1778758}{160721917} a^{5} + \frac{1579568}{160721917} a^{4} - \frac{1468791}{160721917} a^{3} - \frac{465548}{3737719} a^{2} - \frac{1750342}{3737719} a - \frac{172170}{3737719}$, $\frac{1}{1015981587288244933} a^{22} - \frac{1123737}{12778517454919} a^{13} - \frac{101219}{12778517454919} a^{12} + \frac{208017}{297174824533} a^{11} + \frac{1550794}{297174824533} a^{10} + \frac{1488680}{297174824533} a^{9} + \frac{1018211}{6911042431} a^{8} + \frac{194728}{6911042431} a^{7} - \frac{1367826}{6911042431} a^{6} - \frac{1419867}{160721917} a^{5} - \frac{648004}{160721917} a^{4} + \frac{1485520}{160721917} a^{3} - \frac{1021109}{3737719} a^{2} + \frac{493537}{3737719} a + \frac{1237361}{3737719}$, $\frac{1}{1015981587288244933} a^{23} + \frac{1324222}{12778517454919} a^{13} + \frac{1506758}{12778517454919} a^{12} + \frac{1708656}{297174824533} a^{11} - \frac{586237}{297174824533} a^{10} - \frac{1370414}{297174824533} a^{9} + \frac{756703}{6911042431} a^{8} + \frac{1596867}{6911042431} a^{7} - \frac{1442201}{6911042431} a^{6} + \frac{1499766}{160721917} a^{5} + \frac{1613136}{160721917} a^{4} - \frac{379486}{160721917} a^{3} + \frac{461633}{3737719} a^{2} + \frac{892605}{3737719} a - \frac{783085}{3737719}$, $\frac{1}{43687208253394532119} a^{24} + \frac{985279}{12778517454919} a^{13} + \frac{1559169}{12778517454919} a^{12} + \frac{1454139}{297174824533} a^{11} - \frac{817780}{297174824533} a^{10} - \frac{845971}{297174824533} a^{9} - \frac{640542}{6911042431} a^{8} + \frac{144698}{6911042431} a^{7} - \frac{1809076}{6911042431} a^{6} - \frac{1807492}{160721917} a^{5} + \frac{1356807}{160721917} a^{4} - \frac{985630}{160721917} a^{3} + \frac{309547}{3737719} a^{2} + \frac{436894}{3737719} a + \frac{561066}{3737719}$, $\frac{1}{43687208253394532119} a^{25} + \frac{1461372}{12778517454919} a^{13} + \frac{1565110}{12778517454919} a^{12} + \frac{453581}{297174824533} a^{11} - \frac{76053}{297174824533} a^{10} - \frac{1868401}{297174824533} a^{9} + \frac{1442848}{6911042431} a^{8} + \frac{112375}{6911042431} a^{7} + \frac{1113412}{6911042431} a^{6} - \frac{1507839}{160721917} a^{5} + \frac{31297}{160721917} a^{4} + \frac{146030}{160721917} a^{3} + \frac{1117092}{3737719} a^{2} + \frac{839828}{3737719} a - \frac{812417}{3737719}$, $\frac{1}{43687208253394532119} a^{26} + \frac{41826}{297174824533} a^{13} - \frac{1540352}{12778517454919} a^{12} + \frac{627362}{297174824533} a^{11} - \frac{124988}{297174824533} a^{10} + \frac{513495}{297174824533} a^{9} - \frac{1584254}{6911042431} a^{8} + \frac{1516809}{6911042431} a^{7} - \frac{824745}{6911042431} a^{6} - \frac{1062286}{160721917} a^{5} - \frac{1413949}{160721917} a^{4} - \frac{1338530}{160721917} a^{3} + \frac{295393}{3737719} a^{2} - \frac{700540}{3737719} a + \frac{879259}{3737719}$

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

\(\Q(\zeta_{9})^+\), \(\Q(\zeta_{27})^+\)

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 $27$ $27$ $27$ $27$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{3}$ $27$ $27$ $27$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{3}$ $27$ R $27$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{27}$ $27$

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
43Data not computed