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Properties

Label 18.0.16098825206...6527.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,13^{9}\cdot 19^{15}$
Root discriminant $41.94$
Ramified primes $13, 19$
Class number $6$ (GRH)
Class group $[6]$ (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![39464, 123520, 235274, 94987, 216145, 19893, 72400, 993, 21962, -4689, 10956, -3725, 2872, -911, 328, -85, 16, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 16*x^16 - 85*x^15 + 328*x^14 - 911*x^13 + 2872*x^12 - 3725*x^11 + 10956*x^10 - 4689*x^9 + 21962*x^8 + 993*x^7 + 72400*x^6 + 19893*x^5 + 216145*x^4 + 94987*x^3 + 235274*x^2 + 123520*x + 39464)
 
gp: K = bnfinit(x^18 - 2*x^17 + 16*x^16 - 85*x^15 + 328*x^14 - 911*x^13 + 2872*x^12 - 3725*x^11 + 10956*x^10 - 4689*x^9 + 21962*x^8 + 993*x^7 + 72400*x^6 + 19893*x^5 + 216145*x^4 + 94987*x^3 + 235274*x^2 + 123520*x + 39464, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 16 x^{16} - 85 x^{15} + 328 x^{14} - 911 x^{13} + 2872 x^{12} - 3725 x^{11} + 10956 x^{10} - 4689 x^{9} + 21962 x^{8} + 993 x^{7} + 72400 x^{6} + 19893 x^{5} + 216145 x^{4} + 94987 x^{3} + 235274 x^{2} + 123520 x + 39464 \)

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:  \(-160988252069740650830246966527=-\,13^{9}\cdot 19^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.94$
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:  $13, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{1064} a^{15} - \frac{5}{532} a^{14} - \frac{71}{1064} a^{13} - \frac{43}{266} a^{12} - \frac{163}{1064} a^{11} + \frac{3}{266} a^{10} - \frac{225}{1064} a^{9} - \frac{1}{76} a^{8} + \frac{261}{1064} a^{7} + \frac{37}{76} a^{6} - \frac{401}{1064} a^{5} - \frac{5}{28} a^{4} - \frac{261}{1064} a^{3} - \frac{73}{266} a^{2} - \frac{71}{266} a - \frac{58}{133}$, $\frac{1}{241528} a^{16} - \frac{2}{30191} a^{15} - \frac{3203}{241528} a^{14} - \frac{15567}{120764} a^{13} + \frac{33321}{241528} a^{12} - \frac{13869}{120764} a^{11} + \frac{36411}{241528} a^{10} + \frac{10575}{60382} a^{9} - \frac{49663}{241528} a^{8} - \frac{3456}{30191} a^{7} + \frac{117255}{241528} a^{6} + \frac{17711}{60382} a^{5} + \frac{13647}{241528} a^{4} + \frac{6285}{17252} a^{3} - \frac{13848}{30191} a^{2} + \frac{1485}{30191} a - \frac{11489}{30191}$, $\frac{1}{59322183883170821401620812250672715544} a^{17} - \frac{45861278977973770781267003447755}{59322183883170821401620812250672715544} a^{16} + \frac{7529737476922856543827650577915793}{59322183883170821401620812250672715544} a^{15} - \frac{2757739597217417990070495386265116291}{59322183883170821401620812250672715544} a^{14} + \frac{473155144427644624593445355512890365}{8474597697595831628802973178667530792} a^{13} + \frac{13313425646507321953844490955961960353}{59322183883170821401620812250672715544} a^{12} + \frac{13041104428993592149522515511737339009}{59322183883170821401620812250672715544} a^{11} + \frac{11710322450696527155174439990434325381}{59322183883170821401620812250672715544} a^{10} - \frac{8239916537733969170282424008819631019}{59322183883170821401620812250672715544} a^{9} - \frac{1518369789694387403049864389637357647}{8474597697595831628802973178667530792} a^{8} + \frac{8854357571468703871191598041797287459}{59322183883170821401620812250672715544} a^{7} + \frac{25654267330314544434488331828177502229}{59322183883170821401620812250672715544} a^{6} + \frac{10073194500207221543459249315885231147}{59322183883170821401620812250672715544} a^{5} + \frac{15908911535454245927912646413434250895}{59322183883170821401620812250672715544} a^{4} + \frac{730684987184321931006625813371779103}{4237298848797915814401486589333765396} a^{3} - \frac{14095319231543686118035296045630694681}{29661091941585410700810406125336357772} a^{2} + \frac{2469378264440916658632827629936098377}{7415272985396352675202601531334089443} a + \frac{403039995206634621701866301847970995}{7415272985396352675202601531334089443}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( 3170947.06149 \) (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

\(\Q(\sqrt{-247}) \), 3.1.247.1 x3, 3.3.361.1, 6.0.15069223.1, 6.0.5439989503.2 x2, 6.0.5439989503.1, 9.3.1963836210583.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.5439989503.2
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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$13$13.6.3.2$x^{6} - 338 x^{2} + 13182$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.2$x^{6} - 338 x^{2} + 13182$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
13.6.3.2$x^{6} - 338 x^{2} + 13182$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$19.6.5.2$x^{6} - 19$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.2$x^{6} - 19$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.2$x^{6} - 19$$6$$1$$5$$C_6$$[\ ]_{6}$