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Properties

Label 18.0.33020222286...1527.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{9}\cdot 67^{12}$
Root discriminant $43.64$
Ramified primes $7, 67$
Class number $25$ (GRH)
Class group $[5, 5]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16200, -31860, 47648, -41143, 50867, -52920, 42002, -16358, 2812, -5808, 7524, -3044, 100, 210, -2, -52, 32, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 32*x^16 - 52*x^15 - 2*x^14 + 210*x^13 + 100*x^12 - 3044*x^11 + 7524*x^10 - 5808*x^9 + 2812*x^8 - 16358*x^7 + 42002*x^6 - 52920*x^5 + 50867*x^4 - 41143*x^3 + 47648*x^2 - 31860*x + 16200)
 
gp: K = bnfinit(x^18 - 9*x^17 + 32*x^16 - 52*x^15 - 2*x^14 + 210*x^13 + 100*x^12 - 3044*x^11 + 7524*x^10 - 5808*x^9 + 2812*x^8 - 16358*x^7 + 42002*x^6 - 52920*x^5 + 50867*x^4 - 41143*x^3 + 47648*x^2 - 31860*x + 16200, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 32 x^{16} - 52 x^{15} - 2 x^{14} + 210 x^{13} + 100 x^{12} - 3044 x^{11} + 7524 x^{10} - 5808 x^{9} + 2812 x^{8} - 16358 x^{7} + 42002 x^{6} - 52920 x^{5} + 50867 x^{4} - 41143 x^{3} + 47648 x^{2} - 31860 x + 16200 \)

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:  \(-330202222869024796498756781527=-\,7^{9}\cdot 67^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.64$
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:  $7, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{2} + \frac{1}{3} a$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4}$, $\frac{1}{6} a^{13} - \frac{1}{2} a^{6} + \frac{1}{3} a^{5}$, $\frac{1}{630} a^{14} - \frac{1}{90} a^{13} - \frac{29}{630} a^{12} - \frac{5}{63} a^{11} + \frac{13}{210} a^{10} + \frac{22}{315} a^{9} - \frac{71}{210} a^{7} - \frac{43}{630} a^{6} - \frac{29}{90} a^{5} - \frac{8}{315} a^{4} + \frac{173}{630} a^{3} + \frac{2}{105} a^{2} + \frac{146}{315} a + \frac{3}{7}$, $\frac{1}{630} a^{15} + \frac{3}{70} a^{13} - \frac{43}{630} a^{12} + \frac{2}{315} a^{11} + \frac{1}{315} a^{10} - \frac{1}{90} a^{9} + \frac{17}{105} a^{8} - \frac{137}{315} a^{7} - \frac{3}{10} a^{6} + \frac{11}{210} a^{5} + \frac{83}{315} a^{4} + \frac{139}{315} a^{3} + \frac{61}{630} a^{2} + \frac{109}{630} a$, $\frac{1}{2094598650025980} a^{16} - \frac{2}{523649662506495} a^{15} + \frac{296340195629}{523649662506495} a^{14} - \frac{98780065208}{24935698214595} a^{13} - \frac{49430843666969}{1047299325012990} a^{12} + \frac{141920260087}{104729932501299} a^{11} + \frac{31244276200049}{1047299325012990} a^{10} + \frac{7602298574219}{349099775004330} a^{9} - \frac{11596585431925}{209459865002598} a^{8} - \frac{34445810098066}{523649662506495} a^{7} - \frac{38240474787593}{523649662506495} a^{6} - \frac{3099430653923}{9974279285838} a^{5} - \frac{89284027365821}{523649662506495} a^{4} - \frac{1504838308213}{1047299325012990} a^{3} + \frac{849118837607189}{2094598650025980} a^{2} + \frac{10437990419259}{38788863889370} a + \frac{1611715911149}{3878886388937}$, $\frac{1}{1841152213372836420} a^{17} + \frac{431}{1841152213372836420} a^{16} - \frac{58084515769138}{153429351114403035} a^{15} - \frac{48177589194533}{920576106686418210} a^{14} + \frac{15097954490997491}{184115221337283642} a^{13} + \frac{3739389646174871}{92057610668641821} a^{12} - \frac{219030729040906}{7306159576876335} a^{11} - \frac{76323710663930053}{920576106686418210} a^{10} - \frac{968223712973812}{65755436191887015} a^{9} - \frac{136537125094648447}{920576106686418210} a^{8} + \frac{20044634346286597}{102286234076268690} a^{7} + \frac{129468143340572033}{920576106686418210} a^{6} + \frac{35822554288140512}{92057610668641821} a^{5} + \frac{12722176819858724}{460288053343209105} a^{4} - \frac{5919185066453693}{613717404457612140} a^{3} - \frac{55686879450268099}{1841152213372836420} a^{2} + \frac{37632604099397411}{102286234076268690} a + \frac{439581651521828}{1136513711958541}$

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

Class group and class number

$C_{5}\times C_{5}$, which has order $25$ (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:  \( 2912368.54861 \) (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

\(\Q(\sqrt{-7}) \), 3.1.31423.1 x3, 3.3.4489.1, 6.0.6911834503.2, 6.0.1539727.1 x2, 6.0.6911834503.1, 9.3.31027225083967.2 x3

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

Sibling fields

Degree 6 sibling: 6.0.1539727.1
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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$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$$[\ ]_{2}^{3}$
$67$67.9.6.1$x^{9} + 3216 x^{6} + 3443063 x^{3} + 1231925248$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
67.9.6.1$x^{9} + 3216 x^{6} + 3443063 x^{3} + 1231925248$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$