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Properties

Label 18.0.79794198402...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{30}\cdot 5^{9}\cdot 31^{14}$
Root discriminant $403.33$
Ramified primes $2, 3, 5, 31$
Class number $235636128$ (GRH)
Class group $[2, 6, 6, 18, 181818]$ (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![134550076329, -8673549336, 37957823343, -14614241886, 5393042586, -2002698378, 914131653, -144337500, 25952187, -10423648, 518775, 1155498, -151035, -39702, 7908, 738, -138, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 138*x^16 + 738*x^15 + 7908*x^14 - 39702*x^13 - 151035*x^12 + 1155498*x^11 + 518775*x^10 - 10423648*x^9 + 25952187*x^8 - 144337500*x^7 + 914131653*x^6 - 2002698378*x^5 + 5393042586*x^4 - 14614241886*x^3 + 37957823343*x^2 - 8673549336*x + 134550076329)
 
gp: K = bnfinit(x^18 - 6*x^17 - 138*x^16 + 738*x^15 + 7908*x^14 - 39702*x^13 - 151035*x^12 + 1155498*x^11 + 518775*x^10 - 10423648*x^9 + 25952187*x^8 - 144337500*x^7 + 914131653*x^6 - 2002698378*x^5 + 5393042586*x^4 - 14614241886*x^3 + 37957823343*x^2 - 8673549336*x + 134550076329, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 138 x^{16} + 738 x^{15} + 7908 x^{14} - 39702 x^{13} - 151035 x^{12} + 1155498 x^{11} + 518775 x^{10} - 10423648 x^{9} + 25952187 x^{8} - 144337500 x^{7} + 914131653 x^{6} - 2002698378 x^{5} + 5393042586 x^{4} - 14614241886 x^{3} + 37957823343 x^{2} - 8673549336 x + 134550076329 \)

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:  \(-79794198402451666358803683478559912448000000000=-\,2^{18}\cdot 3^{30}\cdot 5^{9}\cdot 31^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $403.33$
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:  $2, 3, 5, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{8} + \frac{1}{3} a^{5}$, $\frac{1}{12} a^{12} - \frac{1}{6} a^{11} + \frac{1}{12} a^{10} + \frac{1}{6} a^{9} - \frac{1}{12} a^{8} + \frac{1}{6} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{12} a^{13} + \frac{1}{12} a^{11} - \frac{1}{12} a^{9} + \frac{5}{12} a^{7} - \frac{1}{6} a^{6} + \frac{5}{12} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{12} a^{14} - \frac{1}{6} a^{11} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} + \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{6} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{24} a^{15} - \frac{1}{24} a^{14} - \frac{1}{24} a^{13} - \frac{1}{24} a^{11} + \frac{1}{12} a^{10} + \frac{3}{8} a^{9} - \frac{1}{6} a^{8} - \frac{1}{24} a^{7} + \frac{1}{12} a^{6} - \frac{3}{8} a^{5} - \frac{1}{8} a^{3} - \frac{1}{8}$, $\frac{1}{1151802795576} a^{16} - \frac{1563812181}{95983566298} a^{15} - \frac{15008529143}{575901397788} a^{14} - \frac{6716069151}{383934265192} a^{13} + \frac{42026896939}{1151802795576} a^{12} + \frac{8649128239}{383934265192} a^{11} + \frac{6435960933}{383934265192} a^{10} + \frac{91291031}{6897022728} a^{9} - \frac{58304894555}{383934265192} a^{8} - \frac{219611200843}{1151802795576} a^{7} - \frac{119276006987}{1151802795576} a^{6} + \frac{110898209035}{1151802795576} a^{5} + \frac{120280743339}{383934265192} a^{4} - \frac{124007196677}{383934265192} a^{3} + \frac{7809411321}{95983566298} a^{2} - \frac{20211655257}{383934265192} a - \frac{44508895425}{383934265192}$, $\frac{1}{314788957500931715219632355187185851526619844641012232511221672565946264} a^{17} - \frac{8200460732264682043891515967047034166953610289171873594047}{104929652500310571739877451729061950508873281547004077503740557521982088} a^{16} - \frac{1318617381303572507334604037433714967938455208440721876170538685481699}{104929652500310571739877451729061950508873281547004077503740557521982088} a^{15} - \frac{6101304536414724270886710183364709096014072669993624327259417184128621}{157394478750465857609816177593592925763309922320506116255610836282973132} a^{14} + \frac{12834923472897911713901963211064956114460728547476337220006031012742183}{314788957500931715219632355187185851526619844641012232511221672565946264} a^{13} - \frac{435072817155710046901880675916134460791217349593194593274658411566719}{52464826250155285869938725864530975254436640773502038751870278760991044} a^{12} + \frac{40446630145636408470625895591182165044657161921624565772854809062332797}{314788957500931715219632355187185851526619844641012232511221672565946264} a^{11} + \frac{2563955170204817906660372654091786734499570690668880709803939890930921}{78697239375232928804908088796796462881654961160253058127805418141486566} a^{10} + \frac{72056340296760722632063697693550995851879371556688180099709215277964123}{314788957500931715219632355187185851526619844641012232511221672565946264} a^{9} + \frac{31014124744121143565781818025040887584557730330876454250311675734755825}{157394478750465857609816177593592925763309922320506116255610836282973132} a^{8} + \frac{15439310006381147854863539959495324463196675725099902340504671933438501}{104929652500310571739877451729061950508873281547004077503740557521982088} a^{7} + \frac{23757931423340013486402597356615669780670780651915357024257608859273621}{78697239375232928804908088796796462881654961160253058127805418141486566} a^{6} - \frac{20954359960746441435860655969133323206746026166418889486179141414500255}{314788957500931715219632355187185851526619844641012232511221672565946264} a^{5} + \frac{629670920341343431500830161844444169094491209468050509570047805756248}{13116206562538821467484681466132743813609160193375509687967569690247761} a^{4} - \frac{2845661799491241532882277982468350693016046819421090668834119697414157}{26232413125077642934969362932265487627218320386751019375935139380495522} a^{3} + \frac{35708824343444286093529709590290375167259082796623225721802747329692491}{104929652500310571739877451729061950508873281547004077503740557521982088} a^{2} - \frac{8288355286713619024032479652029219342346379165743209904837338069305511}{26232413125077642934969362932265487627218320386751019375935139380495522} a + \frac{22764707549006822436488537962757236146729585199056157754165596900791989}{52464826250155285869938725864530975254436640773502038751870278760991044}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{6}\times C_{18}\times C_{181818}$, which has order $235636128$ (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:  \( 54921872.620261565 \) (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

\(\Q(\sqrt{-5}) \), 3.3.77841.2, 3.3.837.1, 6.0.48473770248000.4, 6.0.5604552000.4, 9.9.394775941205626677.2

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 R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.9.15.12$x^{9} + 3 x^{8} + 3 x^{7} + 6 x^{3} + 6$$9$$1$$15$$S_3\times C_3$$[3/2, 2]_{2}$
3.9.15.12$x^{9} + 3 x^{8} + 3 x^{7} + 6 x^{3} + 6$$9$$1$$15$$S_3\times C_3$$[3/2, 2]_{2}$
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$31$31.6.4.2$x^{6} - 31 x^{3} + 11532$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
31.12.10.3$x^{12} - 7471 x^{6} + 19927296$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$