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Properties

Label 18.0.31239274635...6423.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 7^{15}\cdot 13^{12}$
Root discriminant $121.07$
Ramified primes $3, 7, 13$
Class number $1561287$ (GRH)
Class group $[7, 223041]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![134217728, 671416320, 1296838656, 1197861632, 521121744, 56153808, -37749104, -11455716, 3409803, 1925200, 27669, -158904, -22610, 6144, 1674, -132, -57, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 57*x^16 - 132*x^15 + 1674*x^14 + 6144*x^13 - 22610*x^12 - 158904*x^11 + 27669*x^10 + 1925200*x^9 + 3409803*x^8 - 11455716*x^7 - 37749104*x^6 + 56153808*x^5 + 521121744*x^4 + 1197861632*x^3 + 1296838656*x^2 + 671416320*x + 134217728)
 
gp: K = bnfinit(x^18 - 57*x^16 - 132*x^15 + 1674*x^14 + 6144*x^13 - 22610*x^12 - 158904*x^11 + 27669*x^10 + 1925200*x^9 + 3409803*x^8 - 11455716*x^7 - 37749104*x^6 + 56153808*x^5 + 521121744*x^4 + 1197861632*x^3 + 1296838656*x^2 + 671416320*x + 134217728, 1)
 

Normalized defining polynomial

\( x^{18} - 57 x^{16} - 132 x^{15} + 1674 x^{14} + 6144 x^{13} - 22610 x^{12} - 158904 x^{11} + 27669 x^{10} + 1925200 x^{9} + 3409803 x^{8} - 11455716 x^{7} - 37749104 x^{6} + 56153808 x^{5} + 521121744 x^{4} + 1197861632 x^{3} + 1296838656 x^{2} + 671416320 x + 134217728 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-31239274635791169267446408746759536423=-\,3^{24}\cdot 7^{15}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $121.07$
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$$\chi_{819}(1,·)$, $\chi_{819}(451,·)$, $\chi_{819}(646,·)$, $\chi_{819}(523,·)$, $\chi_{819}(781,·)$, $\chi_{819}(139,·)$, $\chi_{819}(22,·)$, $\chi_{819}(601,·)$, $\chi_{819}(94,·)$, $\chi_{819}(289,·)$, $\chi_{819}(802,·)$, $\chi_{819}(484,·)$, $\chi_{819}(40,·)$, $\chi_{819}(430,·)$, $\chi_{819}(445,·)$, $\chi_{819}(625,·)$, $\chi_{819}(118,·)$, $\chi_{819}(61,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{64} a^{7} + \frac{1}{32} a^{5} + \frac{1}{64} a^{3} - \frac{1}{16} a$, $\frac{1}{128} a^{8} + \frac{1}{64} a^{6} + \frac{1}{128} a^{4} - \frac{1}{32} a^{2}$, $\frac{1}{128} a^{9} - \frac{3}{128} a^{5} - \frac{3}{64} a^{3} + \frac{1}{16} a$, $\frac{1}{256} a^{10} - \frac{1}{256} a^{9} - \frac{1}{128} a^{7} - \frac{3}{256} a^{6} - \frac{1}{256} a^{5} - \frac{3}{128} a^{4} + \frac{1}{64} a^{3} + \frac{1}{32} a^{2}$, $\frac{1}{2048} a^{11} + \frac{1}{1024} a^{10} - \frac{7}{2048} a^{9} - \frac{3}{1024} a^{8} + \frac{15}{2048} a^{7} + \frac{23}{1024} a^{6} + \frac{51}{2048} a^{5} + \frac{55}{1024} a^{4} + \frac{113}{512} a^{3} - \frac{19}{256} a^{2} - \frac{1}{4} a$, $\frac{1}{4096} a^{12} - \frac{1}{4096} a^{11} + \frac{3}{4096} a^{10} + \frac{15}{4096} a^{9} - \frac{15}{4096} a^{8} - \frac{31}{4096} a^{7} + \frac{25}{4096} a^{6} + \frac{149}{4096} a^{5} + \frac{117}{2048} a^{4} - \frac{193}{1024} a^{3} - \frac{31}{512} a^{2} + \frac{5}{32} a$, $\frac{1}{8192} a^{13} - \frac{1}{8192} a^{12} - \frac{1}{8192} a^{11} - \frac{9}{8192} a^{10} - \frac{3}{8192} a^{9} - \frac{7}{8192} a^{8} + \frac{61}{8192} a^{7} + \frac{13}{8192} a^{6} - \frac{121}{4096} a^{5} - \frac{23}{2048} a^{4} - \frac{169}{1024} a^{3} + \frac{3}{256} a^{2} + \frac{3}{16} a$, $\frac{1}{8192} a^{14} - \frac{7}{4096} a^{10} - \frac{1}{256} a^{9} + \frac{1}{512} a^{8} + \frac{85}{8192} a^{6} - \frac{13}{256} a^{5} + \frac{11}{1024} a^{4} - \frac{21}{128} a^{3} - \frac{11}{512} a^{2} + \frac{7}{32} a$, $\frac{1}{65536} a^{15} + \frac{1}{16384} a^{12} + \frac{5}{32768} a^{11} - \frac{7}{16384} a^{10} + \frac{17}{8192} a^{9} + \frac{55}{16384} a^{8} - \frac{3}{65536} a^{7} - \frac{381}{16384} a^{6} + \frac{17}{1024} a^{5} - \frac{197}{4096} a^{4} + \frac{563}{4096} a^{3} + \frac{35}{512} a^{2} - \frac{5}{32} a$, $\frac{1}{262144} a^{16} + \frac{1}{262144} a^{15} + \frac{1}{32768} a^{14} - \frac{3}{65536} a^{13} - \frac{1}{131072} a^{12} - \frac{1}{131072} a^{11} + \frac{123}{65536} a^{10} - \frac{155}{65536} a^{9} + \frac{105}{262144} a^{8} - \frac{1991}{262144} a^{7} - \frac{431}{65536} a^{6} + \frac{369}{16384} a^{5} - \frac{93}{8192} a^{4} - \frac{2509}{16384} a^{3} + \frac{1}{64} a^{2} + \frac{9}{64} a$, $\frac{1}{41647374196249319698361243992588288} a^{17} - \frac{211844509504112239280396169}{5205921774531164962295155499073536} a^{16} - \frac{168140360010611219399296030913}{41647374196249319698361243992588288} a^{15} + \frac{360560849610658115275613697451}{10411843549062329924590310998147072} a^{14} - \frac{418563308525781148960262781707}{20823687098124659849180621996294144} a^{13} + \frac{132963828293100207363936025281}{2602960887265582481147577749536768} a^{12} + \frac{2780754816294572897881829808639}{20823687098124659849180621996294144} a^{11} + \frac{5517200827665202911828296539545}{5205921774531164962295155499073536} a^{10} - \frac{28938809363657545136690517786955}{41647374196249319698361243992588288} a^{9} + \frac{12680153072666868599966641476889}{5205921774531164962295155499073536} a^{8} + \frac{114894440634251943964449109760515}{41647374196249319698361243992588288} a^{7} + \frac{319558779280146439270318932667915}{10411843549062329924590310998147072} a^{6} + \frac{154644919106701729438449880818989}{2602960887265582481147577749536768} a^{5} - \frac{30079197324759724000976051371459}{2602960887265582481147577749536768} a^{4} - \frac{112379313611944861720570968708107}{2602960887265582481147577749536768} a^{3} + \frac{3235999295614484708274468483737}{20335631931762363133965451168256} a^{2} - \frac{714815984371219524614145437981}{10167815965881181566982725584128} a + \frac{5190083290041437262787118759}{39718031116723365496026271813}$

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

Class group and class number

$C_{7}\times C_{223041}$, which has order $1561287$ (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:  $8$
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:  \( 12689307150.302711 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-7}) \), 3.3.13689.1, 3.3.3969.1, 3.3.670761.2, 3.3.8281.1, 6.0.64274331303.4, 6.0.110270727.2, 6.0.3149442233847.9, 6.0.480024727.1, 9.9.301789003173921081.3

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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{18}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
3Data not computed
7Data not computed
13Data not computed