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Properties

Label 18.0.75333124892...0191.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{12}\cdot 97^{15}\cdot 127^{9}$
Root discriminant $1866.14$
Ramified primes $7, 97, 127$
Class number $24652633824$ (GRH)
Class group $[3, 3, 3, 3, 42, 84, 86268]$ (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![3741267643296382976, -74967510509588480, -74294080126316288, 16967797388265088, -3926480656027888, -911575249761792, 221396206694440, 20737038801648, -4575731495955, -244055933606, 49548274205, 1510843992, -288098278, -5120388, 927642, 8808, -1527, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 1527*x^16 + 8808*x^15 + 927642*x^14 - 5120388*x^13 - 288098278*x^12 + 1510843992*x^11 + 49548274205*x^10 - 244055933606*x^9 - 4575731495955*x^8 + 20737038801648*x^7 + 221396206694440*x^6 - 911575249761792*x^5 - 3926480656027888*x^4 + 16967797388265088*x^3 - 74294080126316288*x^2 - 74967510509588480*x + 3741267643296382976)
 
gp: K = bnfinit(x^18 - 6*x^17 - 1527*x^16 + 8808*x^15 + 927642*x^14 - 5120388*x^13 - 288098278*x^12 + 1510843992*x^11 + 49548274205*x^10 - 244055933606*x^9 - 4575731495955*x^8 + 20737038801648*x^7 + 221396206694440*x^6 - 911575249761792*x^5 - 3926480656027888*x^4 + 16967797388265088*x^3 - 74294080126316288*x^2 - 74967510509588480*x + 3741267643296382976, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 1527 x^{16} + 8808 x^{15} + 927642 x^{14} - 5120388 x^{13} - 288098278 x^{12} + 1510843992 x^{11} + 49548274205 x^{10} - 244055933606 x^{9} - 4575731495955 x^{8} + 20737038801648 x^{7} + 221396206694440 x^{6} - 911575249761792 x^{5} - 3926480656027888 x^{4} + 16967797388265088 x^{3} - 74294080126316288 x^{2} - 74967510509588480 x + 3741267643296382976 \)

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:  \(-75333124892092297262992244259187556332425903757023354560191=-\,7^{12}\cdot 97^{15}\cdot 127^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1866.14$
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:  $7, 97, 127$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{16} a^{7} + \frac{3}{16} a^{3} - \frac{1}{4} a$, $\frac{1}{128} a^{8} - \frac{1}{32} a^{7} + \frac{1}{64} a^{6} - \frac{1}{16} a^{5} + \frac{1}{128} a^{4} - \frac{1}{32} a^{3} - \frac{1}{32} a^{2} + \frac{1}{8} a$, $\frac{1}{12416} a^{9} - \frac{3}{12416} a^{8} - \frac{93}{6208} a^{7} + \frac{177}{6208} a^{6} + \frac{609}{12416} a^{5} - \frac{11}{12416} a^{4} - \frac{63}{1552} a^{3} + \frac{107}{3104} a^{2} - \frac{179}{776} a - \frac{15}{97}$, $\frac{1}{24832} a^{10} - \frac{1}{24832} a^{9} + \frac{1}{12416} a^{8} - \frac{9}{12416} a^{7} - \frac{623}{24832} a^{6} + \frac{431}{24832} a^{5} - \frac{277}{6208} a^{4} + \frac{825}{6208} a^{3} + \frac{5}{388} a^{2} - \frac{42}{97} a - \frac{15}{97}$, $\frac{1}{49664} a^{11} - \frac{1}{49664} a^{9} - \frac{5}{24832} a^{8} - \frac{269}{49664} a^{7} + \frac{163}{12416} a^{6} + \frac{2761}{49664} a^{5} + \frac{331}{24832} a^{4} + \frac{1545}{12416} a^{3} - \frac{1411}{6208} a^{2} - \frac{471}{1552} a$, $\frac{1}{149488640} a^{12} + \frac{15}{1868608} a^{11} - \frac{331}{29897728} a^{10} - \frac{235}{7474432} a^{9} + \frac{46803}{149488640} a^{8} + \frac{25185}{3737216} a^{7} - \frac{476899}{21355520} a^{6} + \frac{379573}{7474432} a^{5} - \frac{142131}{18686080} a^{4} - \frac{168531}{934304} a^{3} + \frac{270129}{1334720} a^{2} + \frac{769}{1552} a - \frac{1138}{20855}$, $\frac{1}{298977280} a^{13} - \frac{1}{298977280} a^{12} - \frac{213}{59795456} a^{11} + \frac{475}{59795456} a^{10} + \frac{5613}{298977280} a^{9} - \frac{17663}{298977280} a^{8} + \frac{189279}{6952960} a^{7} - \frac{4520787}{298977280} a^{6} - \frac{48137}{1541120} a^{5} - \frac{3466413}{74744320} a^{4} + \frac{3339571}{37372160} a^{3} + \frac{134689}{667360} a^{2} - \frac{69949}{333680} a + \frac{3754}{20855}$, $\frac{1}{4185681920} a^{14} - \frac{1}{597954560} a^{13} + \frac{1}{837136384} a^{12} - \frac{543}{837136384} a^{11} - \frac{83957}{4185681920} a^{10} + \frac{137099}{4185681920} a^{9} - \frac{7863593}{4185681920} a^{8} + \frac{90078311}{4185681920} a^{7} - \frac{1045501}{523210240} a^{6} + \frac{1010949}{18686080} a^{5} + \frac{2849617}{65401280} a^{4} - \frac{4466971}{261605120} a^{3} - \frac{1124523}{9343040} a^{2} - \frac{65717}{291970} a - \frac{7582}{29197}$, $\frac{1}{8371363840} a^{15} + \frac{3}{2092840960} a^{13} + \frac{1}{1046420480} a^{12} - \frac{36221}{4185681920} a^{11} - \frac{3481}{209284096} a^{10} + \frac{58787}{2092840960} a^{9} + \frac{1177373}{1046420480} a^{8} + \frac{1497393}{86302720} a^{7} + \frac{824391}{149488640} a^{6} + \frac{67583}{2696960} a^{5} + \frac{20028171}{523210240} a^{4} - \frac{332257}{10677760} a^{3} - \frac{3568947}{18686080} a^{2} + \frac{596213}{2335760} a + \frac{117}{485}$, $\frac{1}{42673734271366065437692801251422044160} a^{16} - \frac{198803848893110141544369189}{5334216783920758179711600156427755520} a^{15} + \frac{128481910544332923862752051}{10668433567841516359423200312855511040} a^{14} - \frac{288642137428525086056957003}{1066843356784151635942320031285551104} a^{13} + \frac{36160028756133241364983564103}{21336867135683032718846400625711022080} a^{12} - \frac{17499349993247531673852208771137}{5334216783920758179711600156427755520} a^{11} + \frac{2499441295835427319161717510967}{248103106228872473474958146810593280} a^{10} + \frac{80342775957488430886410167444287}{5334216783920758179711600156427755520} a^{9} + \frac{1930951697415702071326168885487727}{6096247753052295062527543035917434880} a^{8} - \frac{33450868284002672904506742519395179}{2667108391960379089855800078213877760} a^{7} - \frac{78686218147701517436815476510670981}{5334216783920758179711600156427755520} a^{6} - \frac{67043670481498572351243120104188103}{1333554195980189544927900039106938880} a^{5} + \frac{2567670244400994861016581087532029}{2667108391960379089855800078213877760} a^{4} - \frac{565789501774667117931557071326791}{6803847938674436453713775709729280} a^{3} - \frac{2180131197676233762075708083074701}{23813467785360527587998214984052480} a^{2} - \frac{74781874620401780158218154388351}{297668347317006594849977687300656} a - \frac{2244156761692316113239437973633}{13288765505223508698659718183065}$, $\frac{1}{38480186145702382441421371864314458748112090793246720} a^{17} + \frac{6773933599143}{7696037229140476488284274372862891749622418158649344} a^{16} + \frac{197521963237404213896673145301736223030393}{9620046536425595610355342966078614687028022698311680} a^{15} + \frac{365105214990545648314468841901078412334811}{9620046536425595610355342966078614687028022698311680} a^{14} + \frac{11589896983628603895893430223136292863678259}{19240093072851191220710685932157229374056045396623360} a^{13} - \frac{49931379415713071451989163629169186021511223}{19240093072851191220710685932157229374056045396623360} a^{12} - \frac{3515529030730047701636056336264725274223112753}{9620046536425595610355342966078614687028022698311680} a^{11} - \frac{75716865180819761699686103856114830289342116707}{9620046536425595610355342966078614687028022698311680} a^{10} - \frac{10399508522838448037169164873380809239842797519}{38480186145702382441421371864314458748112090793246720} a^{9} + \frac{9312171532458376547830428754887957302320660658039}{7696037229140476488284274372862891749622418158649344} a^{8} - \frac{39456501145760273029481176618107920157169945733141}{4810023268212797805177671483039307343514011349155840} a^{7} - \frac{62233801705541542750443046104934328325286077697023}{4810023268212797805177671483039307343514011349155840} a^{6} - \frac{110666137155975582046228126742263251873839329498553}{2405011634106398902588835741519653671757005674577920} a^{5} + \frac{27809171173707271058488200581983639006318572246035}{481002326821279780517767148303930734351401134915584} a^{4} + \frac{4992079536893178587923435109847212566606616537887}{42946636323328551831943495384279529852803672760320} a^{3} + \frac{740037749273467038764211963275221028048921481399}{3067616880237753702281678241734252132343119482880} a^{2} - \frac{18089458931520224710070094409511701635777700693}{268416477020803448949646846151747061580022954752} a + \frac{25516951284807992964821931315570101697771147}{1497859804803590674942225703971802798995663810}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{42}\times C_{84}\times C_{86268}$, which has order $24652633824$ (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:  \( 141934307507722030 \) (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

\(\Q(\sqrt{-12319}) \), 3.1.12319.1 x3, 3.3.461041.2, 6.0.1869503857759.1, Deg 6, Deg 6 x2, 9.3.19471595670631104050933071.1 x3

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

Sibling fields

Degree 6 sibling: data not computed
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 ${\href{/LocalNumberField/2.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$7$7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
$97$97.6.5.5$x^{6} + 12125$$6$$1$$5$$C_6$$[\ ]_{6}$
97.6.5.5$x^{6} + 12125$$6$$1$$5$$C_6$$[\ ]_{6}$
97.6.5.5$x^{6} + 12125$$6$$1$$5$$C_6$$[\ ]_{6}$
$127$127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$
127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$