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Properties

Label 18.0.20093016350...1152.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{9}\cdot 7^{10}\cdot 13^{10}$
Root discriminant $42.46$
Ramified primes $2, 3, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times S_3^2$ (as 18T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4096, -22528, 44288, -33024, -2176, 16256, -9680, 1616, 4088, -3800, 2032, -536, 42, 118, -7, -9, 10, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 10*x^16 - 9*x^15 - 7*x^14 + 118*x^13 + 42*x^12 - 536*x^11 + 2032*x^10 - 3800*x^9 + 4088*x^8 + 1616*x^7 - 9680*x^6 + 16256*x^5 - 2176*x^4 - 33024*x^3 + 44288*x^2 - 22528*x + 4096)
 
gp: K = bnfinit(x^18 - 3*x^17 + 10*x^16 - 9*x^15 - 7*x^14 + 118*x^13 + 42*x^12 - 536*x^11 + 2032*x^10 - 3800*x^9 + 4088*x^8 + 1616*x^7 - 9680*x^6 + 16256*x^5 - 2176*x^4 - 33024*x^3 + 44288*x^2 - 22528*x + 4096, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 10 x^{16} - 9 x^{15} - 7 x^{14} + 118 x^{13} + 42 x^{12} - 536 x^{11} + 2032 x^{10} - 3800 x^{9} + 4088 x^{8} + 1616 x^{7} - 9680 x^{6} + 16256 x^{5} - 2176 x^{4} - 33024 x^{3} + 44288 x^{2} - 22528 x + 4096 \)

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:  \(-200930163501792205662554161152=-\,2^{18}\cdot 3^{9}\cdot 7^{10}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.46$
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:  $2, 3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{8} a^{9} + \frac{1}{16} a^{8} - \frac{1}{16} a^{7} + \frac{1}{8} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{12} - \frac{1}{32} a^{11} - \frac{1}{16} a^{10} + \frac{1}{32} a^{9} + \frac{3}{32} a^{8} + \frac{1}{16} a^{7} + \frac{1}{8} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{11} + \frac{1}{32} a^{10} - \frac{1}{8} a^{9} - \frac{1}{32} a^{8} + \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{64} a^{14} - \frac{1}{64} a^{13} - \frac{1}{64} a^{11} - \frac{1}{64} a^{10} + \frac{1}{16} a^{9} - \frac{3}{32} a^{8} + \frac{1}{16} a^{7} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} - \frac{3}{8} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{256} a^{15} + \frac{1}{256} a^{14} + \frac{1}{128} a^{13} + \frac{3}{256} a^{12} - \frac{3}{256} a^{11} + \frac{3}{128} a^{10} - \frac{5}{128} a^{9} + \frac{1}{32} a^{8} + \frac{1}{32} a^{7} + \frac{5}{32} a^{6} - \frac{1}{32} a^{5} - \frac{3}{16} a^{4} + \frac{5}{16} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{123392} a^{16} - \frac{5}{123392} a^{15} - \frac{109}{30848} a^{14} - \frac{713}{123392} a^{13} - \frac{277}{123392} a^{12} - \frac{245}{15424} a^{11} - \frac{2343}{61696} a^{10} + \frac{2001}{30848} a^{9} + \frac{737}{15424} a^{8} - \frac{277}{15424} a^{7} + \frac{2393}{15424} a^{6} - \frac{26}{241} a^{5} - \frac{849}{7712} a^{4} + \frac{181}{3856} a^{3} - \frac{311}{1928} a^{2} + \frac{145}{964} a - \frac{93}{241}$, $\frac{1}{1595097256131705417728} a^{17} - \frac{5745934389888023}{1595097256131705417728} a^{16} - \frac{338866157822196133}{797548628065852708864} a^{15} - \frac{11178721661408929825}{1595097256131705417728} a^{14} + \frac{1240527586970439949}{1595097256131705417728} a^{13} - \frac{11740516134773797239}{797548628065852708864} a^{12} - \frac{15427278045288186127}{797548628065852708864} a^{11} + \frac{1221817796739001113}{24923394627057897152} a^{10} - \frac{6353422662875259869}{99693578508231588608} a^{9} + \frac{11348394877719079933}{199387157016463177216} a^{8} - \frac{23460504317989992389}{199387157016463177216} a^{7} + \frac{20447905638111043111}{99693578508231588608} a^{6} - \frac{10844700312775261945}{99693578508231588608} a^{5} + \frac{202285349746690811}{24923394627057897152} a^{4} - \frac{158821267982499477}{1780242473361278368} a^{3} + \frac{2315297286839889265}{6230848656764474288} a^{2} + \frac{257017861846691353}{6230848656764474288} a - \frac{76308943797031663}{1557712164191118572}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( -\frac{33791063290749}{3853273527406592} a^{17} + \frac{83163500008499}{3853273527406592} a^{16} - \frac{146255895426307}{1926636763703296} a^{15} + \frac{145556417080237}{3853273527406592} a^{14} + \frac{319135754802959}{3853273527406592} a^{13} - \frac{1906863789803913}{1926636763703296} a^{12} - \frac{1740131002464181}{1926636763703296} a^{11} + \frac{1018760688311745}{240829595462912} a^{10} - \frac{3735376769984855}{240829595462912} a^{9} + \frac{12000190567123727}{481659190925824} a^{8} - \frac{10690311906789047}{481659190925824} a^{7} - \frac{6344304316505175}{240829595462912} a^{6} + \frac{17048333286350749}{240829595462912} a^{5} - \frac{6245044816945993}{60207398865728} a^{4} - \frac{1135028298646069}{30103699432864} a^{3} + \frac{4071134319790741}{15051849716432} a^{2} - \frac{3625977850620357}{15051849716432} a + \frac{243921580809807}{3762962429108} \) (order $6$)
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:  \( 297602831.7716659 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.728.1, 6.0.14309568.1, 6.0.223587.2

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

Sibling fields

Degree 12 sibling: data not computed
Degree 18 siblings: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{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}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.1$x^{6} - 28$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$13$13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.3.1$x^{6} - 52 x^{4} + 676 x^{2} - 79092$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$