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Properties

Label 18.0.10842057567...6848.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{9}\cdot 7^{12}\cdot 19^{15}$
Root discriminant $147.44$
Ramified primes $2, 3, 7, 19$
Class number $1508416$ (GRH)
Class group $[2, 2, 2, 2, 14, 6734]$ (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![1562155624, -994714992, 1429338096, -638296924, 562163766, -244964646, 165055169, -62180514, 31604088, -9597594, 4133913, -1061466, 331962, -46410, 11382, -866, 174, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 174*x^16 - 866*x^15 + 11382*x^14 - 46410*x^13 + 331962*x^12 - 1061466*x^11 + 4133913*x^10 - 9597594*x^9 + 31604088*x^8 - 62180514*x^7 + 165055169*x^6 - 244964646*x^5 + 562163766*x^4 - 638296924*x^3 + 1429338096*x^2 - 994714992*x + 1562155624)
 
gp: K = bnfinit(x^18 - 6*x^17 + 174*x^16 - 866*x^15 + 11382*x^14 - 46410*x^13 + 331962*x^12 - 1061466*x^11 + 4133913*x^10 - 9597594*x^9 + 31604088*x^8 - 62180514*x^7 + 165055169*x^6 - 244964646*x^5 + 562163766*x^4 - 638296924*x^3 + 1429338096*x^2 - 994714992*x + 1562155624, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 174 x^{16} - 866 x^{15} + 11382 x^{14} - 46410 x^{13} + 331962 x^{12} - 1061466 x^{11} + 4133913 x^{10} - 9597594 x^{9} + 31604088 x^{8} - 62180514 x^{7} + 165055169 x^{6} - 244964646 x^{5} + 562163766 x^{4} - 638296924 x^{3} + 1429338096 x^{2} - 994714992 x + 1562155624 \)

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:  \(-1084205756727498980519837945309853646848=-\,2^{18}\cdot 3^{9}\cdot 7^{12}\cdot 19^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $147.44$
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:  $2, 3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1596=2^{2}\cdot 3\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{1596}(1,·)$, $\chi_{1596}(863,·)$, $\chi_{1596}(683,·)$, $\chi_{1596}(961,·)$, $\chi_{1596}(457,·)$, $\chi_{1596}(407,·)$, $\chi_{1596}(911,·)$, $\chi_{1596}(121,·)$, $\chi_{1596}(277,·)$, $\chi_{1596}(1367,·)$, $\chi_{1596}(1369,·)$, $\chi_{1596}(1247,·)$, $\chi_{1596}(107,·)$, $\chi_{1596}(1261,·)$, $\chi_{1596}(179,·)$, $\chi_{1596}(1033,·)$, $\chi_{1596}(505,·)$, $\chi_{1596}(1019,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{8} - \frac{1}{3} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{9} + \frac{1}{6} a^{5} + \frac{1}{6} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{6} - \frac{1}{3} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{7} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{126} a^{12} - \frac{2}{63} a^{11} - \frac{5}{126} a^{10} + \frac{1}{21} a^{9} + \frac{1}{126} a^{8} + \frac{10}{63} a^{7} + \frac{22}{63} a^{5} + \frac{61}{126} a^{4} - \frac{23}{63} a^{3} + \frac{10}{63} a^{2} + \frac{2}{9} a - \frac{4}{63}$, $\frac{1}{126} a^{13} + \frac{1}{18} a^{10} + \frac{2}{63} a^{9} + \frac{1}{42} a^{8} + \frac{17}{126} a^{7} - \frac{19}{126} a^{6} + \frac{3}{14} a^{5} + \frac{5}{21} a^{4} - \frac{17}{126} a^{3} - \frac{13}{42} a^{2} + \frac{31}{63} a - \frac{16}{63}$, $\frac{1}{882} a^{14} - \frac{1}{882} a^{12} + \frac{16}{441} a^{11} + \frac{17}{294} a^{10} - \frac{5}{98} a^{9} + \frac{37}{882} a^{8} - \frac{10}{147} a^{7} + \frac{3}{98} a^{6} - \frac{47}{126} a^{5} + \frac{37}{294} a^{4} + \frac{37}{126} a^{3} + \frac{1}{6} a^{2} - \frac{15}{49} a + \frac{193}{441}$, $\frac{1}{882} a^{15} - \frac{1}{882} a^{13} - \frac{1}{294} a^{12} + \frac{22}{441} a^{11} - \frac{17}{882} a^{10} - \frac{13}{441} a^{9} + \frac{26}{441} a^{8} + \frac{31}{441} a^{7} + \frac{8}{63} a^{6} - \frac{253}{882} a^{5} + \frac{13}{63} a^{4} + \frac{31}{63} a^{3} + \frac{59}{882} a^{2} + \frac{16}{49} a - \frac{1}{63}$, $\frac{1}{71916821858441684218471524} a^{16} + \frac{2177411344632308857361}{5993068488203473684872627} a^{15} - \frac{388001840691768624493}{733845121004506981821138} a^{14} - \frac{292993873564768915309}{11986136976406947369745254} a^{13} + \frac{45921178901197731828665}{35958410929220842109235762} a^{12} - \frac{173920945159917399534991}{5136915847031548872747966} a^{11} + \frac{1037692505820399962826049}{35958410929220842109235762} a^{10} - \frac{1160322738940907573515619}{17979205464610421054617881} a^{9} - \frac{424305908363801832762853}{71916821858441684218471524} a^{8} - \frac{56723548041251300008439}{856152641171924812124661} a^{7} + \frac{38040111902587258083728}{17979205464610421054617881} a^{6} - \frac{59653041821360612173594}{1997689496067824561624209} a^{5} + \frac{5439167415552634088178149}{23972273952813894739490508} a^{4} - \frac{484875403923686066478659}{11986136976406947369745254} a^{3} - \frac{62938357939871592022195}{17979205464610421054617881} a^{2} - \frac{8638068897900497994283795}{17979205464610421054617881} a + \frac{101895774897100200758555}{17979205464610421054617881}$, $\frac{1}{16497066771886882260050047395744112433844} a^{17} + \frac{32610767178209}{8248533385943441130025023697872056216922} a^{16} - \frac{246448926256864375822651711298740501}{458251854774635618334723538770669789829} a^{15} + \frac{3096312920190158018788088870753856661}{8248533385943441130025023697872056216922} a^{14} + \frac{3971095781017456548487926794232500801}{2749511128647813710008341232624018738974} a^{13} - \frac{25128785099851621316338815881351600851}{8248533385943441130025023697872056216922} a^{12} + \frac{38321884833639739224045170624259421730}{1374755564323906855004170616312009369487} a^{11} + \frac{97499128299735909418537376863785225404}{1374755564323906855004170616312009369487} a^{10} - \frac{93149696805750980725366032307452669455}{1833007419098542473338894155082679159316} a^{9} - \frac{94943504181227928979158384783407997139}{4124266692971720565012511848936028108461} a^{8} + \frac{404457684431741183862852414099152097512}{4124266692971720565012511848936028108461} a^{7} + \frac{1936114981369041225042334085467708232}{65464550682090802619246219824381398547} a^{6} + \frac{5946794632588264750596301098665525883719}{16497066771886882260050047395744112433844} a^{5} - \frac{2721330889026201298499723608372618915963}{8248533385943441130025023697872056216922} a^{4} + \frac{903243793005619348462668607999583039900}{4124266692971720565012511848936028108461} a^{3} - \frac{3585704067800397540100947603675080618015}{8248533385943441130025023697872056216922} a^{2} + \frac{357366019463745397407217815255173670416}{1374755564323906855004170616312009369487} a - \frac{689912019568382023561424156341648722617}{4124266692971720565012511848936028108461}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{14}\times C_{6734}$, which has order $1508416$ (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:  \( 833965.2438558349 \) (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

\(\Q(\sqrt{-57}) \), 3.3.361.1, \(\Q(\zeta_{7})^+\), 3.3.17689.1, 3.3.17689.2, 6.0.4278699072.1, 6.0.28457497152.2, 6.0.10273156471872.1, 6.0.10273156471872.2, 9.9.5534900853769.1

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
19Data not computed