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Properties

Label 18.0.92493227793...6048.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 7^{6}\cdot 19^{14}\cdot 73^{14}$
Root discriminant $1461.45$
Ramified primes $2, 3, 7, 19, 73$
Class number $23300706240$ (GRH)
Class group $[3, 3, 3, 12, 12, 5992980]$ (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![1628413597910449, 2126907556454464, 1861722334970505, 969083882090814, 380485121195032, 96784777040119, 19375023447318, 2290282965516, 336726026476, 28542470943, 3988265085, 225068368, 28464247, 956039, 149163, 2513, 470, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 470*x^16 + 2513*x^15 + 149163*x^14 + 956039*x^13 + 28464247*x^12 + 225068368*x^11 + 3988265085*x^10 + 28542470943*x^9 + 336726026476*x^8 + 2290282965516*x^7 + 19375023447318*x^6 + 96784777040119*x^5 + 380485121195032*x^4 + 969083882090814*x^3 + 1861722334970505*x^2 + 2126907556454464*x + 1628413597910449)
 
gp: K = bnfinit(x^18 - x^17 + 470*x^16 + 2513*x^15 + 149163*x^14 + 956039*x^13 + 28464247*x^12 + 225068368*x^11 + 3988265085*x^10 + 28542470943*x^9 + 336726026476*x^8 + 2290282965516*x^7 + 19375023447318*x^6 + 96784777040119*x^5 + 380485121195032*x^4 + 969083882090814*x^3 + 1861722334970505*x^2 + 2126907556454464*x + 1628413597910449, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 470 x^{16} + 2513 x^{15} + 149163 x^{14} + 956039 x^{13} + 28464247 x^{12} + 225068368 x^{11} + 3988265085 x^{10} + 28542470943 x^{9} + 336726026476 x^{8} + 2290282965516 x^{7} + 19375023447318 x^{6} + 96784777040119 x^{5} + 380485121195032 x^{4} + 969083882090814 x^{3} + 1861722334970505 x^{2} + 2126907556454464 x + 1628413597910449 \)

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:  \(-924932277939078761663470646803127775979456162566125826048=-\,2^{12}\cdot 3^{9}\cdot 7^{6}\cdot 19^{14}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1461.45$
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:  $2, 3, 7, 19, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{7} a^{4} - \frac{1}{7} a^{3} + \frac{1}{7} a^{2}$, $\frac{1}{7} a^{5} + \frac{1}{7} a^{2}$, $\frac{1}{49} a^{6} - \frac{1}{49} a^{5} + \frac{1}{49} a^{4} - \frac{1}{7} a^{3} - \frac{3}{7} a^{2}$, $\frac{1}{343} a^{7} - \frac{1}{343} a^{6} - \frac{20}{343} a^{5} + \frac{2}{49} a^{4} + \frac{8}{49} a^{3} - \frac{3}{7} a^{2} + \frac{1}{7} a$, $\frac{1}{2401} a^{8} - \frac{1}{2401} a^{7} - \frac{20}{2401} a^{6} - \frac{12}{343} a^{5} + \frac{22}{343} a^{4} + \frac{23}{49} a^{3} + \frac{15}{49} a^{2} + \frac{3}{7} a$, $\frac{1}{16807} a^{9} - \frac{1}{16807} a^{8} - \frac{20}{16807} a^{7} - \frac{12}{2401} a^{6} + \frac{169}{2401} a^{5} + \frac{2}{343} a^{4} - \frac{160}{343} a^{3} - \frac{18}{49} a^{2} + \frac{3}{7} a$, $\frac{1}{117649} a^{10} - \frac{1}{117649} a^{9} - \frac{20}{117649} a^{8} - \frac{12}{16807} a^{7} + \frac{169}{16807} a^{6} - \frac{145}{2401} a^{5} - \frac{13}{2401} a^{4} + \frac{10}{343} a^{3} + \frac{10}{49} a^{2} + \frac{3}{7} a$, $\frac{1}{823543} a^{11} - \frac{1}{823543} a^{10} - \frac{20}{823543} a^{9} - \frac{12}{117649} a^{8} + \frac{169}{117649} a^{7} - \frac{145}{16807} a^{6} - \frac{13}{16807} a^{5} + \frac{10}{2401} a^{4} + \frac{59}{343} a^{3} + \frac{3}{49} a^{2} + \frac{1}{7} a$, $\frac{1}{40353607} a^{12} - \frac{8}{40353607} a^{11} + \frac{134}{40353607} a^{10} - \frac{13}{5764801} a^{9} + \frac{176}{5764801} a^{8} + \frac{71}{823543} a^{7} + \frac{5286}{823543} a^{6} + \frac{8150}{117649} a^{5} - \frac{571}{16807} a^{4} - \frac{985}{2401} a^{3} - \frac{34}{343} a^{2}$, $\frac{1}{282475249} a^{13} - \frac{1}{282475249} a^{12} + \frac{78}{282475249} a^{11} + \frac{121}{40353607} a^{10} + \frac{85}{40353607} a^{9} + \frac{247}{5764801} a^{8} + \frac{5783}{5764801} a^{7} - \frac{3371}{823543} a^{6} + \frac{2777}{117649} a^{5} - \frac{870}{16807} a^{4} - \frac{480}{2401} a^{3} - \frac{34}{343} a^{2} - \frac{3}{7} a$, $\frac{1}{1977326743} a^{14} - \frac{1}{1977326743} a^{13} - \frac{20}{1977326743} a^{12} - \frac{110}{282475249} a^{11} + \frac{953}{282475249} a^{10} + \frac{1066}{40353607} a^{9} + \frac{575}{40353607} a^{8} + \frac{45}{5764801} a^{7} + \frac{5190}{823543} a^{6} + \frac{4971}{117649} a^{5} + \frac{844}{16807} a^{4} + \frac{1005}{2401} a^{3} + \frac{145}{343} a^{2} + \frac{2}{7} a$, $\frac{1}{38663176157734020697} a^{15} + \frac{8373925097}{38663176157734020697} a^{14} + \frac{11263916553}{38663176157734020697} a^{13} - \frac{46875616705}{5523310879676288671} a^{12} - \frac{1911060024103}{5523310879676288671} a^{11} - \frac{780653273457}{789044411382326953} a^{10} - \frac{2618399918784}{789044411382326953} a^{9} + \frac{25498061984}{2300421024438271} a^{8} - \frac{1995390249812}{16102947171067897} a^{7} + \frac{1070493604067}{328631574919753} a^{6} - \frac{20288423543516}{328631574919753} a^{5} + \frac{110143633484}{46947367845679} a^{4} - \frac{2272530995965}{6706766835097} a^{3} - \frac{195102538249}{958109547871} a^{2} - \frac{6012943872}{19553256079} a - \frac{1279970415}{2793322297}$, $\frac{1}{1894495631728967014153} a^{16} - \frac{1}{1894495631728967014153} a^{15} + \frac{10288961511}{99710296406787737587} a^{14} - \frac{438460332483}{270642233104138144879} a^{13} + \frac{258167209238}{270642233104138144879} a^{12} + \frac{13651512599020}{38663176157734020697} a^{11} - \frac{163664007506446}{38663176157734020697} a^{10} - \frac{85931893679573}{5523310879676288671} a^{9} - \frac{28294533773584}{789044411382326953} a^{8} + \frac{20473041184544}{16102947171067897} a^{7} - \frac{6458425721723}{847523535319363} a^{6} + \frac{126007018278277}{2300421024438271} a^{5} + \frac{4095007287561}{328631574919753} a^{4} - \frac{1092912731721}{6706766835097} a^{3} + \frac{60094341203}{958109547871} a^{2} + \frac{6062634819}{19553256079} a + \frac{21}{1387}$, $\frac{1}{13924871211742891053306741196143217} a^{17} + \frac{2623873960574}{13924871211742891053306741196143217} a^{16} + \frac{34667518413347}{13924871211742891053306741196143217} a^{15} - \frac{254645065530578408515009}{1989267315963270150472391599449031} a^{14} + \frac{3045056851620810805371888}{1989267315963270150472391599449031} a^{13} + \frac{928282796823490708963008}{284181045137610021496055942778433} a^{12} - \frac{90039358678250709660018778}{284181045137610021496055942778433} a^{11} + \frac{117572410357079592019737859}{40597292162515717356579420396919} a^{10} + \frac{4309199251432420617627489}{446124089697974916006367257109} a^{9} - \frac{20745866157706931081986352}{118359452368850487920056619233} a^{8} + \frac{90781678298691115716085874}{118359452368850487920056619233} a^{7} - \frac{123728438115107877950713840}{16908493195550069702865231319} a^{6} + \frac{29140914719055446399903378}{2415499027935724243266461617} a^{5} - \frac{387802685164659531768770}{7042271218471499251505719} a^{4} + \frac{1914246476032024954813764}{7042271218471499251505719} a^{3} - \frac{53703094906544622302755}{143719820785132637785831} a^{2} + \frac{178256194129637639196}{1579338689946512503141} a - \frac{63538904080008903560}{419008223863360460017}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{12}\times C_{12}\times C_{5992980}$, which has order $23300706240$ (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:  \( -\frac{13319239110400848292}{13924871211742891053306741196143217} a^{17} + \frac{29052704303968118105}{13924871211742891053306741196143217} a^{16} - \frac{6309400774842547927576}{13924871211742891053306741196143217} a^{15} - \frac{3709691104306704833014}{1989267315963270150472391599449031} a^{14} - \frac{280493621779928397210506}{1989267315963270150472391599449031} a^{13} - \frac{212971409842228980341223}{284181045137610021496055942778433} a^{12} - \frac{7532313985012072650047860}{284181045137610021496055942778433} a^{11} - \frac{7497470681830641414398890}{40597292162515717356579420396919} a^{10} - \frac{85188039977429656976830}{23480215247261837684545645111} a^{9} - \frac{2746657417912094045051269}{118359452368850487920056619233} a^{8} - \frac{35387490237272892539865132}{118359452368850487920056619233} a^{7} - \frac{31490333246084419364566316}{16908493195550069702865231319} a^{6} - \frac{40346163786633045564687236}{2415499027935724243266461617} a^{5} - \frac{3695089768295661070354781}{49295898529300494760540033} a^{4} - \frac{2088104312050943692329246}{7042271218471499251505719} a^{3} - \frac{94641865678157899438226}{143719820785132637785831} a^{2} - \frac{2318856507603891152580}{1579338689946512503141} a - \frac{1455755149046857715}{2126945298798784061} \) (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:  \( 1270819096089.6284 \) (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

\(\Q(\sqrt{-3}) \), 3.3.38836.1, 3.3.1923769.2, 6.0.40722342192.1, 6.0.99923953464747.4, 9.9.216775063213574541708241216.1

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$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$$[\ ]_{2}^{3}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
$19$19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.6.5.1$x^{6} - 304$$6$$1$$5$$C_6$$[\ ]_{6}$
19.6.5.1$x^{6} - 304$$6$$1$$5$$C_6$$[\ ]_{6}$
$73$73.3.2.3$x^{3} - 1825$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.3$x^{3} - 1825$$3$$1$$2$$C_3$$[\ ]_{3}$
73.6.5.6$x^{6} + 228125$$6$$1$$5$$C_6$$[\ ]_{6}$
73.6.5.6$x^{6} + 228125$$6$$1$$5$$C_6$$[\ ]_{6}$