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Properties

Label 18.0.34460307596...4064.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 7^{14}\cdot 17^{9}$
Root discriminant $107.12$
Ramified primes $2, 3, 7, 17$
Class number $235872$ (GRH)
Class group $[2, 36, 3276]$ (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![3038757625, -3133190175, 2252365185, -552590982, 284543364, -9702642, 8452852, -3211662, 2206947, -1769347, 813441, -286986, 106762, -21096, 5532, -716, 123, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 123*x^16 - 716*x^15 + 5532*x^14 - 21096*x^13 + 106762*x^12 - 286986*x^11 + 813441*x^10 - 1769347*x^9 + 2206947*x^8 - 3211662*x^7 + 8452852*x^6 - 9702642*x^5 + 284543364*x^4 - 552590982*x^3 + 2252365185*x^2 - 3133190175*x + 3038757625)
 
gp: K = bnfinit(x^18 - 9*x^17 + 123*x^16 - 716*x^15 + 5532*x^14 - 21096*x^13 + 106762*x^12 - 286986*x^11 + 813441*x^10 - 1769347*x^9 + 2206947*x^8 - 3211662*x^7 + 8452852*x^6 - 9702642*x^5 + 284543364*x^4 - 552590982*x^3 + 2252365185*x^2 - 3133190175*x + 3038757625, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 123 x^{16} - 716 x^{15} + 5532 x^{14} - 21096 x^{13} + 106762 x^{12} - 286986 x^{11} + 813441 x^{10} - 1769347 x^{9} + 2206947 x^{8} - 3211662 x^{7} + 8452852 x^{6} - 9702642 x^{5} + 284543364 x^{4} - 552590982 x^{3} + 2252365185 x^{2} - 3133190175 x + 3038757625 \)

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:  \(-3446030759610882930198241186568024064=-\,2^{12}\cdot 3^{21}\cdot 7^{14}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $107.12$
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:  $2, 3, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{6} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{7} - \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{116} a^{15} - \frac{2}{29} a^{14} + \frac{7}{116} a^{13} + \frac{1}{58} a^{12} + \frac{7}{29} a^{11} + \frac{1}{29} a^{10} - \frac{7}{116} a^{9} + \frac{1}{58} a^{8} - \frac{2}{29} a^{7} - \frac{11}{58} a^{6} + \frac{3}{116} a^{5} - \frac{25}{58} a^{4} + \frac{53}{116} a^{3} - \frac{12}{29} a^{2} - \frac{8}{29} a + \frac{23}{58}$, $\frac{1}{7540} a^{16} + \frac{1}{7540} a^{15} - \frac{877}{7540} a^{14} - \frac{243}{3770} a^{13} - \frac{209}{3770} a^{12} - \frac{1571}{7540} a^{11} - \frac{7}{260} a^{10} + \frac{229}{7540} a^{9} - \frac{227}{3770} a^{8} - \frac{1747}{7540} a^{7} - \frac{1123}{7540} a^{6} - \frac{2227}{7540} a^{5} - \frac{3123}{7540} a^{4} + \frac{622}{1885} a^{3} + \frac{1}{10} a^{2} - \frac{677}{7540} a - \frac{75}{377}$, $\frac{1}{1039735892301611581793961155098762878275145966498995868939118395515204300} a^{17} - \frac{10992387541054289242824125863995846517469744357425109148757279025699}{1039735892301611581793961155098762878275145966498995868939118395515204300} a^{16} - \frac{215777161040917479460398729583881392934430619302392805309118068272423}{259933973075402895448490288774690719568786491624748967234779598878801075} a^{15} - \frac{5472426525833188499642263634458687226035648597322052852585065109422684}{259933973075402895448490288774690719568786491624748967234779598878801075} a^{14} + \frac{40557691734882052345621007491527941330283113789180957042877711287659211}{519867946150805790896980577549381439137572983249497934469559197757602150} a^{13} + \frac{28032181389130052737963913659046901166538809844099743859906523110284431}{259933973075402895448490288774690719568786491624748967234779598878801075} a^{12} - \frac{108797783413446732687988865473184806006650621796334579792094703151318799}{519867946150805790896980577549381439137572983249497934469559197757602150} a^{11} + \frac{18807034285487336741363800059987260598955508516619308974530885398032146}{259933973075402895448490288774690719568786491624748967234779598878801075} a^{10} + \frac{205073113938697659736466706603120925600625745411539292945918875329300081}{1039735892301611581793961155098762878275145966498995868939118395515204300} a^{9} + \frac{186781838930061925999231776443677904227291804588409883338253941022138063}{1039735892301611581793961155098762878275145966498995868939118395515204300} a^{8} - \frac{31307222348568501396557830088816824778805888541317626353176521856479749}{519867946150805790896980577549381439137572983249497934469559197757602150} a^{7} + \frac{9155045476386414074224441374015189780252797603441646269258744734750977}{259933973075402895448490288774690719568786491624748967234779598878801075} a^{6} + \frac{1386508939164272358781570048031384748144565053011249080980884143001727}{8963240450875961912016906509472093778234016952577550594302744788924175} a^{5} - \frac{210547138678609534408922648365615669193175132237651279462394447782693211}{519867946150805790896980577549381439137572983249497934469559197757602150} a^{4} + \frac{99157257144947182317221353023768404441759323173880456619819703487684747}{519867946150805790896980577549381439137572983249497934469559197757602150} a^{3} + \frac{100380263732258042512390575071511209085540201406295145646009280534414802}{259933973075402895448490288774690719568786491624748967234779598878801075} a^{2} - \frac{70813725194723055301877868515671334023683503314616879140939025026999957}{207947178460322316358792231019752575655029193299799173787823679103040860} a + \frac{2807155119946772457649986300879741771026783671347651037810119941600975}{41589435692064463271758446203950515131005838659959834757564735820608172}$

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

Class group and class number

$C_{2}\times C_{36}\times C_{3276}$, which has order $235872$ (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:  \( 296124.35954857944 \) (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

\(\Q(\sqrt{-51}) \), 3.3.756.1, \(\Q(\zeta_{7})^+\), 6.0.8423869104.4, 6.0.318495051.1, 9.9.1037426999616.1

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$17$17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.2$x^{6} - 289 x^{2} + 14739$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$