↑

Properties

Label 18.0.23956678900...2384.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{31}\cdot 7^{9}\cdot 31^{15}$
Root discriminant $487.24$
Ramified primes $2, 3, 7, 31$
Class number $1078583688$ (GRH)
Class group $[3, 1638, 219492]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![18360337249, 80259221724, 156763071546, 183224962614, 145837576599, 85555837932, 38939538201, 14164032300, 4180850511, 1007188198, 196786164, 30551556, 3597450, 280182, 7527, -1410, -159, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 159*x^16 - 1410*x^15 + 7527*x^14 + 280182*x^13 + 3597450*x^12 + 30551556*x^11 + 196786164*x^10 + 1007188198*x^9 + 4180850511*x^8 + 14164032300*x^7 + 38939538201*x^6 + 85555837932*x^5 + 145837576599*x^4 + 183224962614*x^3 + 156763071546*x^2 + 80259221724*x + 18360337249)
 
gp: K = bnfinit(x^18 - 6*x^17 - 159*x^16 - 1410*x^15 + 7527*x^14 + 280182*x^13 + 3597450*x^12 + 30551556*x^11 + 196786164*x^10 + 1007188198*x^9 + 4180850511*x^8 + 14164032300*x^7 + 38939538201*x^6 + 85555837932*x^5 + 145837576599*x^4 + 183224962614*x^3 + 156763071546*x^2 + 80259221724*x + 18360337249, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 159 x^{16} - 1410 x^{15} + 7527 x^{14} + 280182 x^{13} + 3597450 x^{12} + 30551556 x^{11} + 196786164 x^{10} + 1007188198 x^{9} + 4180850511 x^{8} + 14164032300 x^{7} + 38939538201 x^{6} + 85555837932 x^{5} + 145837576599 x^{4} + 183224962614 x^{3} + 156763071546 x^{2} + 80259221724 x + 18360337249 \)

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:  \(-2395667890070146411269283915935172526863463952384=-\,2^{12}\cdot 3^{31}\cdot 7^{9}\cdot 31^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $487.24$
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:  $2, 3, 7, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{54} a^{9} - \frac{1}{18} a^{8} - \frac{1}{18} a^{7} + \frac{1}{9} a^{6} + \frac{1}{9} a^{5} - \frac{7}{18} a^{4} + \frac{1}{18} a^{3} - \frac{1}{18} a^{2} + \frac{4}{9} a - \frac{1}{54}$, $\frac{1}{54} a^{10} + \frac{1}{9} a^{8} - \frac{1}{18} a^{7} + \frac{1}{9} a^{6} - \frac{1}{18} a^{5} - \frac{4}{9} a^{4} + \frac{4}{9} a^{3} - \frac{1}{18} a^{2} - \frac{19}{54} a - \frac{1}{18}$, $\frac{1}{54} a^{11} - \frac{1}{18} a^{8} + \frac{1}{9} a^{7} - \frac{1}{18} a^{6} - \frac{1}{9} a^{5} - \frac{2}{9} a^{4} - \frac{7}{18} a^{3} + \frac{17}{54} a^{2} - \frac{7}{18} a + \frac{4}{9}$, $\frac{1}{378} a^{12} + \frac{1}{189} a^{11} + \frac{1}{189} a^{10} - \frac{1}{126} a^{9} + \frac{2}{63} a^{8} - \frac{5}{126} a^{7} + \frac{1}{21} a^{6} + \frac{4}{63} a^{5} - \frac{37}{126} a^{4} + \frac{149}{378} a^{3} - \frac{47}{378} a^{2} - \frac{37}{189} a + \frac{19}{63}$, $\frac{1}{378} a^{13} - \frac{1}{189} a^{11} - \frac{1}{126} a^{9} - \frac{10}{63} a^{8} - \frac{2}{21} a^{7} + \frac{5}{63} a^{6} - \frac{1}{7} a^{5} + \frac{10}{27} a^{4} - \frac{19}{63} a^{3} - \frac{32}{189} a^{2} - \frac{41}{126} a - \frac{17}{63}$, $\frac{1}{1134} a^{14} + \frac{1}{1134} a^{13} + \frac{1}{1134} a^{12} - \frac{5}{567} a^{11} - \frac{2}{567} a^{10} + \frac{5}{1134} a^{9} + \frac{29}{378} a^{8} + \frac{11}{378} a^{7} + \frac{19}{189} a^{6} - \frac{31}{1134} a^{5} - \frac{227}{567} a^{4} - \frac{65}{567} a^{3} + \frac{179}{567} a^{2} + \frac{158}{567} a + \frac{113}{567}$, $\frac{1}{175770} a^{15} - \frac{2}{29295} a^{14} + \frac{2}{1953} a^{13} - \frac{43}{35154} a^{12} + \frac{143}{58590} a^{11} + \frac{127}{19530} a^{10} - \frac{25}{35154} a^{9} + \frac{956}{9765} a^{8} - \frac{64}{465} a^{7} - \frac{1091}{12555} a^{6} + \frac{6961}{58590} a^{5} + \frac{1159}{19530} a^{4} - \frac{3988}{17577} a^{3} + \frac{968}{4185} a^{2} - \frac{7807}{19530} a - \frac{69841}{175770}$, $\frac{1}{60992190} a^{16} - \frac{1}{6099219} a^{15} - \frac{10307}{30496095} a^{14} - \frac{73}{64542} a^{13} + \frac{14027}{30496095} a^{12} + \frac{245978}{30496095} a^{11} - \frac{209642}{30496095} a^{10} - \frac{81442}{10165365} a^{9} - \frac{105556}{10165365} a^{8} + \frac{279047}{60992190} a^{7} + \frac{717691}{6099219} a^{6} - \frac{630742}{4356585} a^{5} - \frac{330994}{1452195} a^{4} + \frac{30291761}{60992190} a^{3} - \frac{26320351}{60992190} a^{2} - \frac{17048327}{60992190} a + \frac{479957}{1129485}$, $\frac{1}{2165121697208209866075985766032188976110} a^{17} - \frac{13590314431952738068447905776231}{2165121697208209866075985766032188976110} a^{16} - \frac{15372886342577792688388458837814}{10310103320039094600361836981105661791} a^{15} - \frac{569608219884522478991782469040055709}{2165121697208209866075985766032188976110} a^{14} + \frac{2708842759548445865228957675766353359}{2165121697208209866075985766032188976110} a^{13} + \frac{440639600128491812705798140578625727}{360853616201368311012664294338698162685} a^{12} - \frac{2615757892940003185069455244933075169}{2165121697208209866075985766032188976110} a^{11} + \frac{12663582603383295536774861125532049199}{2165121697208209866075985766032188976110} a^{10} + \frac{208034134838858252954002345702988361}{26729897496397652667604762543607271310} a^{9} - \frac{13090553554225413752293727625347923679}{433024339441641973215197153206437795222} a^{8} + \frac{28845058693588158298982858722123346425}{433024339441641973215197153206437795222} a^{7} + \frac{52460644888878294097150045879523334697}{721707232402736622025328588677396325370} a^{6} + \frac{18217138475676789664680725820480066497}{2165121697208209866075985766032188976110} a^{5} - \frac{297936030751788758119846471931812603421}{1082560848604104933037992883016094488055} a^{4} - \frac{71309227769683295415865726740396522782}{360853616201368311012664294338698162685} a^{3} - \frac{826352135702338241609603676038936119237}{2165121697208209866075985766032188976110} a^{2} + \frac{505482567894329802452384529402365060153}{2165121697208209866075985766032188976110} a + \frac{2439106590524380315525972027924606217}{40094846244596479001407143815410906965}$

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

Class group and class number

$C_{3}\times C_{1638}\times C_{219492}$, which has order $1078583688$ (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:  \( 52651214.56199736 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-651}) \), 3.1.23436.2 x3, 3.3.77841.2, 6.0.357559208496.1, Deg 6 x2, 6.0.193283099642619.2, 9.3.8666121461345916813504.1 x3

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 6 sibling: data not computed
Degree 9 sibling: data not computed

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
2Data not computed
3Data not computed
$7$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$31$31.6.5.2$x^{6} - 1519$$6$$1$$5$$C_6$$[\ ]_{6}$
31.6.5.2$x^{6} - 1519$$6$$1$$5$$C_6$$[\ ]_{6}$
31.6.5.2$x^{6} - 1519$$6$$1$$5$$C_6$$[\ ]_{6}$