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Properties

Label 18.0.29543127065...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{45}\cdot 5^{12}$
Root discriminant $72.36$
Ramified primes $2, 3, 5$
Class number $27$ (GRH)
Class group $[3, 3, 3]$ (GRH)
Galois group $He_3:C_2$ (as 18T21)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![154736383, -1368000, -17377920, -1375800, 5724000, -1699200, 1154967, -864000, 193320, -22700, 39600, -14400, 1401, 0, 360, -60, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 60*x^15 + 360*x^14 + 1401*x^12 - 14400*x^11 + 39600*x^10 - 22700*x^9 + 193320*x^8 - 864000*x^7 + 1154967*x^6 - 1699200*x^5 + 5724000*x^4 - 1375800*x^3 - 17377920*x^2 - 1368000*x + 154736383)
 
gp: K = bnfinit(x^18 - 60*x^15 + 360*x^14 + 1401*x^12 - 14400*x^11 + 39600*x^10 - 22700*x^9 + 193320*x^8 - 864000*x^7 + 1154967*x^6 - 1699200*x^5 + 5724000*x^4 - 1375800*x^3 - 17377920*x^2 - 1368000*x + 154736383, 1)
 

Normalized defining polynomial

\( x^{18} - 60 x^{15} + 360 x^{14} + 1401 x^{12} - 14400 x^{11} + 39600 x^{10} - 22700 x^{9} + 193320 x^{8} - 864000 x^{7} + 1154967 x^{6} - 1699200 x^{5} + 5724000 x^{4} - 1375800 x^{3} - 17377920 x^{2} - 1368000 x + 154736383 \)

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:  \(-2954312706550833698643000000000000=-\,2^{12}\cdot 3^{45}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $72.36$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{6} - \frac{1}{4} a^{3} + \frac{1}{4}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{7} - \frac{1}{4} a^{4} + \frac{1}{4} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{5} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{4}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{3}{8} a - \frac{1}{8}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{12} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{3}{8} a^{2} - \frac{1}{8}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{12} - \frac{1}{8} a^{3} + \frac{1}{8}$, $\frac{1}{544} a^{16} - \frac{9}{544} a^{15} + \frac{13}{272} a^{14} - \frac{3}{544} a^{13} + \frac{45}{544} a^{12} + \frac{1}{17} a^{11} + \frac{33}{272} a^{10} - \frac{13}{272} a^{9} - \frac{27}{136} a^{8} + \frac{3}{16} a^{7} - \frac{41}{272} a^{6} - \frac{15}{34} a^{5} - \frac{75}{544} a^{4} + \frac{123}{544} a^{3} + \frac{97}{272} a^{2} + \frac{101}{544} a - \frac{131}{544}$, $\frac{1}{952559377809178586555447179036987089537358105256580087328} a^{17} - \frac{67602154338685034273045935166743670125835504643593813}{119069922226147323319430897379623386192169763157072510916} a^{16} + \frac{11428735139592285490463808291206535044656931112865491641}{952559377809178586555447179036987089537358105256580087328} a^{15} + \frac{3488250207483465263486047683760929390758162955482005621}{86596307073561689686858834457907917230668918659689098848} a^{14} - \frac{470607841466354106752070425391499589346752395268291965}{10133610402225304112291991266350926484439979843155107312} a^{13} + \frac{1479996586100837920595664205121363230394870337495981795}{20267220804450608224583982532701852968879959686310214624} a^{12} + \frac{34905653119911103980999775059719990071006131813493159733}{476279688904589293277723589518493544768679052628290043664} a^{11} + \frac{13687699953871360763656998618960708073982909528539906677}{119069922226147323319430897379623386192169763157072510916} a^{10} - \frac{29261158268359345751551860769080467063541573846382139795}{476279688904589293277723589518493544768679052628290043664} a^{9} - \frac{104458214339684516157855478837685856368865583296509852975}{476279688904589293277723589518493544768679052628290043664} a^{8} - \frac{10166266861755693459860478507576128956259441419944382463}{238139844452294646638861794759246772384339526314145021832} a^{7} - \frac{1415474790193711895121242545420754954842776106445171003}{43298153536780844843429417228953958615334459329844549424} a^{6} - \frac{30492078258316944715438114846460534686714436394202783305}{86596307073561689686858834457907917230668918659689098848} a^{5} + \frac{6439226006373277336916261066562735981048122493022487803}{59534961113073661659715448689811693096084881578536255458} a^{4} - \frac{303608052254464115853872746505750282697902118370484810411}{952559377809178586555447179036987089537358105256580087328} a^{3} - \frac{362658617910196893631468462633694913519474347353193033713}{952559377809178586555447179036987089537358105256580087328} a^{2} - \frac{16606904835056608951151697921859880347974331993220662253}{43298153536780844843429417228953958615334459329844549424} a - \frac{260819299889511490459586466898194242972910684952018954035}{952559377809178586555447179036987089537358105256580087328}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}$, which has order $27$ (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:  \( -\frac{554948221951815701311845}{320337888452293344375532388759204} a^{17} - \frac{1289299981219261135261875}{320337888452293344375532388759204} a^{16} + \frac{28395842220448784122880943}{640675776904586688751064777518408} a^{15} + \frac{1243080370279106688683250}{7280406555733939644898463380891} a^{14} - \frac{749398661118378839991375}{6815699754304113710117710399132} a^{13} - \frac{51635826650044393799586795}{13631399508608227420235420798264} a^{12} + \frac{1056594571224592342531303035}{160168944226146672187766194379602} a^{11} + \frac{2132949576824631231368110875}{80084472113073336093883097189801} a^{10} + \frac{12287158953877453382328459110}{80084472113073336093883097189801} a^{9} - \frac{69516582167709602369580655125}{160168944226146672187766194379602} a^{8} + \frac{27325011592146951381592002375}{80084472113073336093883097189801} a^{7} - \frac{12104191629796403713577824245}{7280406555733939644898463380891} a^{6} + \frac{482335166395694904958466138745}{29121626222935758579593853523564} a^{5} - \frac{2987905764178361809689187933125}{320337888452293344375532388759204} a^{4} + \frac{9341911755521796718328782367145}{640675776904586688751064777518408} a^{3} - \frac{24403981316586238198809305436375}{160168944226146672187766194379602} a^{2} + \frac{5796599693678855749155733756875}{29121626222935758579593853523564} a + \frac{477286481144931339749813504849269}{640675776904586688751064777518408} \) (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:  \( 597898480.2407053 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$He_3:C_2$ (as 18T21):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $He_3:C_2$
Character table for $He_3:C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.243.1 x3, 6.0.177147.2, 9.1.31381059609000000.8 x3

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

Sibling fields

Degree 9 siblings: data not computed
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 ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$$[\ ]_{3}^{6}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
3Data not computed
$5$5.6.4.2$x^{6} - 5 x^{3} + 50$$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$$[\ ]_{3}^{6}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$