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Properties

Label 18.0.47844481638...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{9}\cdot 5^{9}\cdot 7^{15}$
Root discriminant $39.20$
Ramified primes $2, 3, 5, 7$
Class number $104$ (GRH)
Class group $[2, 2, 26]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![580021, -163192, 686430, -268108, 174674, -121254, 36962, -11994, -11207, -2378, -1155, 1544, 928, -262, 137, -92, 18, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 18*x^16 - 92*x^15 + 137*x^14 - 262*x^13 + 928*x^12 + 1544*x^11 - 1155*x^10 - 2378*x^9 - 11207*x^8 - 11994*x^7 + 36962*x^6 - 121254*x^5 + 174674*x^4 - 268108*x^3 + 686430*x^2 - 163192*x + 580021)
 
gp: K = bnfinit(x^18 - 4*x^17 + 18*x^16 - 92*x^15 + 137*x^14 - 262*x^13 + 928*x^12 + 1544*x^11 - 1155*x^10 - 2378*x^9 - 11207*x^8 - 11994*x^7 + 36962*x^6 - 121254*x^5 + 174674*x^4 - 268108*x^3 + 686430*x^2 - 163192*x + 580021, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 18 x^{16} - 92 x^{15} + 137 x^{14} - 262 x^{13} + 928 x^{12} + 1544 x^{11} - 1155 x^{10} - 2378 x^{9} - 11207 x^{8} - 11994 x^{7} + 36962 x^{6} - 121254 x^{5} + 174674 x^{4} - 268108 x^{3} + 686430 x^{2} - 163192 x + 580021 \)

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:  \(-47844481638506531328000000000=-\,2^{18}\cdot 3^{9}\cdot 5^{9}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.20$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{381} a^{15} + \frac{19}{381} a^{14} - \frac{7}{127} a^{12} + \frac{158}{381} a^{11} - \frac{9}{127} a^{10} - \frac{37}{127} a^{9} - \frac{29}{127} a^{8} + \frac{104}{381} a^{7} - \frac{5}{127} a^{6} - \frac{173}{381} a^{5} + \frac{188}{381} a^{4} + \frac{56}{381} a^{3} - \frac{17}{381} a^{2} + \frac{140}{381} a - \frac{70}{381}$, $\frac{1}{381} a^{16} + \frac{20}{381} a^{14} - \frac{7}{127} a^{13} + \frac{49}{381} a^{12} + \frac{19}{381} a^{11} + \frac{7}{127} a^{10} - \frac{10}{381} a^{9} + \frac{106}{381} a^{8} + \frac{41}{381} a^{7} - \frac{142}{381} a^{6} + \frac{173}{381} a^{5} + \frac{40}{381} a^{4} + \frac{62}{381} a^{3} + \frac{82}{381} a^{2} - \frac{190}{381} a + \frac{20}{127}$, $\frac{1}{41571761648623811621586311772042452561831080683} a^{17} - \frac{777182898751780197703081527248367740282492}{3197827819124908586275870136310957889371621591} a^{16} - \frac{17387889731289211059973373848589456141405842}{13857253882874603873862103924014150853943693561} a^{15} - \frac{1611924722510832249346803547351626558874876804}{13857253882874603873862103924014150853943693561} a^{14} - \frac{3583265243952092672521140699449654464020828631}{41571761648623811621586311772042452561831080683} a^{13} - \frac{2552726884171287764963283142378002269543237414}{41571761648623811621586311772042452561831080683} a^{12} + \frac{2362047336780186507855936324772627300471450780}{13857253882874603873862103924014150853943693561} a^{11} - \frac{4807285323730473394280224690749776046707728207}{13857253882874603873862103924014150853943693561} a^{10} - \frac{119709077527974827244400166299645932652236200}{1065942606374969528758623378770319296457207197} a^{9} - \frac{1569324660055895016303770376896447801815736729}{41571761648623811621586311772042452561831080683} a^{8} - \frac{5437662499326713372978567092861734140691016663}{13857253882874603873862103924014150853943693561} a^{7} - \frac{1159035463719940694645067093156327154767851774}{41571761648623811621586311772042452561831080683} a^{6} + \frac{4739517852299154594484151210970858947376197545}{41571761648623811621586311772042452561831080683} a^{5} + \frac{3815983126033087143527735949408298224776066870}{13857253882874603873862103924014150853943693561} a^{4} + \frac{10488061189036928287953953152910175415049003507}{41571761648623811621586311772042452561831080683} a^{3} - \frac{991513630909314192292533800141469656633249632}{3197827819124908586275870136310957889371621591} a^{2} - \frac{19709276297728597751799149296956178445450105684}{41571761648623811621586311772042452561831080683} a + \frac{97057169602910444727458388152418813873544831}{3197827819124908586275870136310957889371621591}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{26}$, which has order $104$ (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:  \( 90069.70392878134 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-105}) \), \(\Q(\zeta_{7})^+\), 3.1.980.1, 6.0.3630312000.2, 6.0.3630312000.1, 9.3.941192000.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 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 R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$$[2]^{3}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7Data not computed