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Properties

Label 18.0.41301528122...6352.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{27}\cdot 7^{9}$
Root discriminant $38.88$
Ramified primes $2, 3, 7$
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![1632457, -6384660, 7131054, -2357414, 1646274, -491256, 147569, -118380, 64794, -19734, 16878, -2472, 1674, -444, 96, -40, 12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 12*x^16 - 40*x^15 + 96*x^14 - 444*x^13 + 1674*x^12 - 2472*x^11 + 16878*x^10 - 19734*x^9 + 64794*x^8 - 118380*x^7 + 147569*x^6 - 491256*x^5 + 1646274*x^4 - 2357414*x^3 + 7131054*x^2 - 6384660*x + 1632457)
 
gp: K = bnfinit(x^18 + 12*x^16 - 40*x^15 + 96*x^14 - 444*x^13 + 1674*x^12 - 2472*x^11 + 16878*x^10 - 19734*x^9 + 64794*x^8 - 118380*x^7 + 147569*x^6 - 491256*x^5 + 1646274*x^4 - 2357414*x^3 + 7131054*x^2 - 6384660*x + 1632457, 1)
 

Normalized defining polynomial

\( x^{18} + 12 x^{16} - 40 x^{15} + 96 x^{14} - 444 x^{13} + 1674 x^{12} - 2472 x^{11} + 16878 x^{10} - 19734 x^{9} + 64794 x^{8} - 118380 x^{7} + 147569 x^{6} - 491256 x^{5} + 1646274 x^{4} - 2357414 x^{3} + 7131054 x^{2} - 6384660 x + 1632457 \)

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:  \(-41301528122037146650360676352=-\,2^{27}\cdot 3^{27}\cdot 7^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.88$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{89} a^{16} - \frac{17}{89} a^{15} + \frac{28}{89} a^{14} + \frac{31}{89} a^{13} + \frac{24}{89} a^{12} + \frac{30}{89} a^{11} + \frac{41}{89} a^{10} + \frac{33}{89} a^{9} - \frac{38}{89} a^{8} + \frac{27}{89} a^{7} + \frac{38}{89} a^{6} - \frac{17}{89} a^{5} - \frac{21}{89} a^{4} + \frac{38}{89} a^{3} - \frac{34}{89} a^{2} + \frac{12}{89} a + \frac{18}{89}$, $\frac{1}{7383054706390234376833451675604630379508032344173083141} a^{17} - \frac{22564949852877845408643770118724286999656967439034940}{7383054706390234376833451675604630379508032344173083141} a^{16} + \frac{3447094102022699202757585329443793933226569862997291094}{7383054706390234376833451675604630379508032344173083141} a^{15} + \frac{1186631576165120216304122336480937552306171651193845193}{7383054706390234376833451675604630379508032344173083141} a^{14} - \frac{973913542433376483812210318271181954830091776032167762}{7383054706390234376833451675604630379508032344173083141} a^{13} - \frac{2865033397762003424807593611260387559966045753420872293}{7383054706390234376833451675604630379508032344173083141} a^{12} - \frac{1012867087043905910002381726810721660296564864459204040}{7383054706390234376833451675604630379508032344173083141} a^{11} - \frac{3593680067311627913097642914519325662283230011172480629}{7383054706390234376833451675604630379508032344173083141} a^{10} + \frac{3121245347744445956492272451957469076384296958941678125}{7383054706390234376833451675604630379508032344173083141} a^{9} - \frac{1404621026085173906602000733340931687936281382523901800}{7383054706390234376833451675604630379508032344173083141} a^{8} + \frac{3352831557639349372385787809509313984556448237781726998}{7383054706390234376833451675604630379508032344173083141} a^{7} - \frac{3609296784007349520306978467832023239558481678711684018}{7383054706390234376833451675604630379508032344173083141} a^{6} + \frac{3292357528970970969385600935501572968466229556445211746}{7383054706390234376833451675604630379508032344173083141} a^{5} - \frac{2946698537012821568788333062972858736132850767080752801}{7383054706390234376833451675604630379508032344173083141} a^{4} - \frac{1630395399687945352839406902914076886585894804720323939}{7383054706390234376833451675604630379508032344173083141} a^{3} + \frac{2756425849993761381310985658177382861070597012727475251}{7383054706390234376833451675604630379508032344173083141} a^{2} - \frac{701452602225919580418603030679545132717165056225023530}{7383054706390234376833451675604630379508032344173083141} a + \frac{3489279697876563075988118119693447591135359744318823914}{7383054706390234376833451675604630379508032344173083141}$

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:  \( 55250.70198920589 \) (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

\(\Q(\sqrt{-42}) \), \(\Q(\zeta_{9})^+\), 3.1.648.1, 6.0.3456649728.1, 6.0.3456649728.10, 9.3.272097792.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.12.18.15$x^{12} - 16 x^{10} + 24 x^{6} + 64 x^{4} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
3Data not computed
$7$7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.12.6.1$x^{12} + 294 x^{8} + 3430 x^{6} + 21609 x^{4} + 487403 x^{2} + 2941225$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$