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Properties

Label 18.0.19554767585...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{33}\cdot 5^{12}\cdot 163^{12}$
Root discriminant $1037.96$
Ramified primes $2, 3, 5, 163$
Class number $26244$ (GRH)
Class group $[3, 3, 3, 3, 18, 18]$ (GRH)
Galois group $C_3\times C_3:S_3$ (as 18T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![57871188, -374881932, 1157395500, -2002296426, 2444612454, -1846066752, 634588399, -21041973, 9141591, -832476, -868809, 294723, -46082, 2259, -489, -18, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 18*x^15 - 489*x^14 + 2259*x^13 - 46082*x^12 + 294723*x^11 - 868809*x^10 - 832476*x^9 + 9141591*x^8 - 21041973*x^7 + 634588399*x^6 - 1846066752*x^5 + 2444612454*x^4 - 2002296426*x^3 + 1157395500*x^2 - 374881932*x + 57871188)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 18*x^15 - 489*x^14 + 2259*x^13 - 46082*x^12 + 294723*x^11 - 868809*x^10 - 832476*x^9 + 9141591*x^8 - 21041973*x^7 + 634588399*x^6 - 1846066752*x^5 + 2444612454*x^4 - 2002296426*x^3 + 1157395500*x^2 - 374881932*x + 57871188, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 18 x^{15} - 489 x^{14} + 2259 x^{13} - 46082 x^{12} + 294723 x^{11} - 868809 x^{10} - 832476 x^{9} + 9141591 x^{8} - 21041973 x^{7} + 634588399 x^{6} - 1846066752 x^{5} + 2444612454 x^{4} - 2002296426 x^{3} + 1157395500 x^{2} - 374881932 x + 57871188 \)

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:  \(-1955476758561888850288369070881021250289363000000000000=-\,2^{12}\cdot 3^{33}\cdot 5^{12}\cdot 163^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1037.96$
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:  $2, 3, 5, 163$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{5}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{9} - \frac{1}{6} a^{6} + \frac{1}{6} a^{3}$, $\frac{1}{342} a^{13} - \frac{17}{342} a^{12} + \frac{23}{171} a^{11} - \frac{17}{114} a^{10} - \frac{7}{114} a^{9} - \frac{7}{57} a^{8} + \frac{11}{342} a^{7} - \frac{73}{342} a^{6} + \frac{79}{171} a^{5} + \frac{17}{114} a^{4} + \frac{53}{114} a^{3} - \frac{16}{57} a^{2} - \frac{7}{19} a$, $\frac{1}{342} a^{14} - \frac{5}{114} a^{12} + \frac{47}{342} a^{11} + \frac{4}{57} a^{10} - \frac{1}{6} a^{9} - \frac{1}{18} a^{8} + \frac{1}{3} a^{7} + \frac{1}{6} a^{6} + \frac{1}{342} a^{5} + \frac{1}{3} a^{4} - \frac{43}{114} a^{3} - \frac{8}{57} a^{2} - \frac{5}{19} a$, $\frac{1}{1930530527910} a^{15} - \frac{142077403}{193053052791} a^{14} - \frac{141424027}{101606869890} a^{13} - \frac{54470793683}{1930530527910} a^{12} + \frac{7447568584}{64351017597} a^{11} - \frac{463145459}{8815207890} a^{10} + \frac{161453387783}{1930530527910} a^{9} + \frac{153857835911}{965265263955} a^{8} + \frac{405892017301}{1930530527910} a^{7} + \frac{73685575699}{386106105582} a^{6} - \frac{3986263939}{107251695995} a^{5} + \frac{187705159217}{643510175970} a^{4} + \frac{146435901382}{321755087985} a^{3} + \frac{8758557529}{21450339199} a^{2} + \frac{28837731329}{107251695995} a + \frac{28407726}{77326385}$, $\frac{1}{18506854820189233800990} a^{16} - \frac{1005698551}{18506854820189233800990} a^{15} + \frac{17906746301142536237}{18506854820189233800990} a^{14} - \frac{503163011044700569}{411263440448649640022} a^{13} - \frac{734282897346226861916}{9253427410094616900495} a^{12} + \frac{631613106707872328069}{18506854820189233800990} a^{11} - \frac{1082812019991434019091}{18506854820189233800990} a^{10} - \frac{9339819947675014141}{56769493313463907365} a^{9} - \frac{8909145887663735802191}{18506854820189233800990} a^{8} - \frac{2368876712849942517797}{6168951606729744600330} a^{7} + \frac{3231946879373366030909}{9253427410094616900495} a^{6} - \frac{485553279644897996153}{974044990536275463210} a^{5} - \frac{430653692977599437578}{1028158601121624100055} a^{4} - \frac{3007919355509255248759}{6168951606729744600330} a^{3} + \frac{842774498583041501197}{3084475803364872300165} a^{2} + \frac{21243545761247267717}{54113610585348636845} a - \frac{8675357268168064}{39014859830820935}$, $\frac{1}{607277090959820063257579866519385230} a^{17} + \frac{5329092690962}{303638545479910031628789933259692615} a^{16} + \frac{73012907348492095817573}{303638545479910031628789933259692615} a^{15} + \frac{7978244342293599138556759407654}{6747523232886889591750887405770947} a^{14} - \frac{42554268768828253162844591444161}{31961952155780003329346308764178170} a^{13} - \frac{13340181699767062721478370123056793}{607277090959820063257579866519385230} a^{12} + \frac{37316431110953361075805989065071037}{303638545479910031628789933259692615} a^{11} - \frac{9497628809224995521406896414188523}{121455418191964012651515973303877046} a^{10} + \frac{63365075421967612796277514777317611}{607277090959820063257579866519385230} a^{9} + \frac{29495673204652998005558758404871852}{101212848493303343876263311086564205} a^{8} - \frac{192290665630923379235992660554060143}{607277090959820063257579866519385230} a^{7} - \frac{1416748067763040706574950864441387}{607277090959820063257579866519385230} a^{6} - \frac{103461821821378524788203976723534057}{607277090959820063257579866519385230} a^{5} + \frac{39352256934774726890067607529251439}{202425696986606687752526622173128410} a^{4} + \frac{18192290037609750233318365438894703}{40485139397321337550505324434625682} a^{3} + \frac{33981232576664044945976528372323489}{101212848493303343876263311086564205} a^{2} + \frac{476729134888792910659723213273101}{1775664008654444629408128264676565} a - \frac{340680308251084839287123463024}{1280219184321877887100308770495}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{18}\times C_{18}$, which has order $26244$ (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:  \( \frac{24921398623776617}{1292465022871357038411315} a^{17} - \frac{23484685646650511}{136048949775932319832770} a^{16} + \frac{106061711367566811}{143607224763484115379035} a^{15} - \frac{139639680644256479}{516986009148542815364526} a^{14} - \frac{24825151132399014223}{2584930045742714076822630} a^{13} + \frac{37076948998947826237}{861643348580904692274210} a^{12} - \frac{2285057096889175725019}{2584930045742714076822630} a^{11} + \frac{14556420704941145007221}{2584930045742714076822630} a^{10} - \frac{14003991285178525320553}{861643348580904692274210} a^{9} - \frac{46872085491917501703749}{2584930045742714076822630} a^{8} + \frac{461745366397073566222087}{2584930045742714076822630} a^{7} - \frac{335686957168797603244451}{861643348580904692274210} a^{6} + \frac{552216233951117157459817}{45349649925310773277590} a^{5} - \frac{15054430257045689532699793}{430821674290452346137105} a^{4} + \frac{36387771993734541661154891}{861643348580904692274210} a^{3} - \frac{4338365191898794239331168}{143607224763484115379035} a^{2} + \frac{2085078308652011258361722}{143607224763484115379035} a - \frac{49920774496919951370}{20707602705621357661} \) (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:  \( 78531702270513380 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_3:S_3$ (as 18T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 18 conjugacy class representatives for $C_3\times C_3:S_3$
Character table for $C_3\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.79707.1 x3, Deg 6, Deg 6, 6.0.1771470000.6, 6.0.19059617547.1

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

Sibling fields

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 R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.6.11.15$x^{6} + 3 x^{3} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.15$x^{6} + 3 x^{3} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.15$x^{6} + 3 x^{3} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
$5$5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$163$163.3.2.1$x^{3} - 163$$3$$1$$2$$C_3$$[\ ]_{3}$
163.3.2.1$x^{3} - 163$$3$$1$$2$$C_3$$[\ ]_{3}$
163.3.2.1$x^{3} - 163$$3$$1$$2$$C_3$$[\ ]_{3}$
163.3.2.1$x^{3} - 163$$3$$1$$2$$C_3$$[\ ]_{3}$
163.3.2.1$x^{3} - 163$$3$$1$$2$$C_3$$[\ ]_{3}$
163.3.2.1$x^{3} - 163$$3$$1$$2$$C_3$$[\ ]_{3}$