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Properties

Label 18.12.4521333535...6171.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,7^{12}\cdot 14731^{9}$
Root discriminant $444.13$
Ramified primes $7, 14731$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T362

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-141347172337, -436493676409, 5836768917, 198118012009, -13016923698, -36042457633, 4197277102, 3380634011, -514147324, -178749505, 32117323, 5480324, -1123935, -95339, 22264, 857, -233, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 233*x^16 + 857*x^15 + 22264*x^14 - 95339*x^13 - 1123935*x^12 + 5480324*x^11 + 32117323*x^10 - 178749505*x^9 - 514147324*x^8 + 3380634011*x^7 + 4197277102*x^6 - 36042457633*x^5 - 13016923698*x^4 + 198118012009*x^3 + 5836768917*x^2 - 436493676409*x - 141347172337)
 
gp: K = bnfinit(x^18 - 3*x^17 - 233*x^16 + 857*x^15 + 22264*x^14 - 95339*x^13 - 1123935*x^12 + 5480324*x^11 + 32117323*x^10 - 178749505*x^9 - 514147324*x^8 + 3380634011*x^7 + 4197277102*x^6 - 36042457633*x^5 - 13016923698*x^4 + 198118012009*x^3 + 5836768917*x^2 - 436493676409*x - 141347172337, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 233 x^{16} + 857 x^{15} + 22264 x^{14} - 95339 x^{13} - 1123935 x^{12} + 5480324 x^{11} + 32117323 x^{10} - 178749505 x^{9} - 514147324 x^{8} + 3380634011 x^{7} + 4197277102 x^{6} - 36042457633 x^{5} - 13016923698 x^{4} + 198118012009 x^{3} + 5836768917 x^{2} - 436493676409 x - 141347172337 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-452133353535905818128970950058166588468589916171=-\,7^{12}\cdot 14731^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $444.13$
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:  $7, 14731$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{6} + \frac{1}{6} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a$, $\frac{1}{6} a^{13} - \frac{1}{2} a^{8} + \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{14} - \frac{1}{12} a^{13} - \frac{1}{12} a^{11} - \frac{1}{12} a^{9} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{12} a^{3} + \frac{1}{3} a^{2} + \frac{1}{6} a - \frac{5}{12}$, $\frac{1}{2120616} a^{15} + \frac{2042}{265077} a^{14} + \frac{9971}{706872} a^{13} + \frac{173461}{2120616} a^{12} - \frac{14389}{706872} a^{11} - \frac{113137}{2120616} a^{10} + \frac{24149}{235624} a^{9} - \frac{380515}{1060308} a^{8} + \frac{300875}{1060308} a^{7} - \frac{320641}{1060308} a^{6} - \frac{218923}{530154} a^{5} + \frac{221275}{706872} a^{4} - \frac{146729}{2120616} a^{3} - \frac{325655}{1060308} a^{2} + \frac{255665}{706872} a - \frac{713651}{2120616}$, $\frac{1}{2120616} a^{16} + \frac{9197}{2120616} a^{14} - \frac{36755}{2120616} a^{13} + \frac{147785}{2120616} a^{12} - \frac{41845}{2120616} a^{11} - \frac{46907}{2120616} a^{10} - \frac{136057}{1060308} a^{9} - \frac{122869}{353436} a^{8} - \frac{1157}{353436} a^{7} - \frac{47818}{265077} a^{6} + \frac{292801}{2120616} a^{5} + \frac{84859}{2120616} a^{4} + \frac{185495}{1060308} a^{3} - \frac{1037369}{2120616} a^{2} - \frac{330899}{2120616} a + \frac{29069}{530154}$, $\frac{1}{6514578863838813666628622722524557848942045607059176300840571368} a^{17} - \frac{631244337741501438406333479911161517255511420379032140767}{6514578863838813666628622722524557848942045607059176300840571368} a^{16} - \frac{185752185074720480084065476505879179014098111358945186563}{1085763143973135611104770453754092974823674267843196050140095228} a^{15} + \frac{8783059773066350187573685850799540093727607283829907027996233}{1085763143973135611104770453754092974823674267843196050140095228} a^{14} - \frac{3039561055536844984804260815252127013719162527421794060267825}{6514578863838813666628622722524557848942045607059176300840571368} a^{13} + \frac{455656627343514710329339255727192438624768729113519382001704113}{6514578863838813666628622722524557848942045607059176300840571368} a^{12} + \frac{402987188980246100999933739070854002265335632315693279959502601}{6514578863838813666628622722524557848942045607059176300840571368} a^{11} - \frac{131691808661437840033663692343098741865420164568715677270460579}{3257289431919406833314311361262278924471022803529588150420285684} a^{10} + \frac{345836620032453323042927812689899030341995382346139745025408177}{6514578863838813666628622722524557848942045607059176300840571368} a^{9} - \frac{1179936840216237542188848121521544887359174596644212766955117411}{3257289431919406833314311361262278924471022803529588150420285684} a^{8} - \frac{31375350075583458603182399984042921310050699966713678954104181}{1628644715959703416657155680631139462235511401764794075210142842} a^{7} + \frac{587985128739695509714862762925896098738977118579695270727451551}{6514578863838813666628622722524557848942045607059176300840571368} a^{6} + \frac{781008515514755433486782751238905128780384667320481699053053559}{1628644715959703416657155680631139462235511401764794075210142842} a^{5} - \frac{454263325462512831598501465467905376442170223274470309225372237}{1085763143973135611104770453754092974823674267843196050140095228} a^{4} - \frac{381711405281078021450412100815003353245520918832888990906970786}{814322357979851708328577840315569731117755700882397037605071421} a^{3} + \frac{1537654211976537394175344158612023331955900033694607328957506343}{3257289431919406833314311361262278924471022803529588150420285684} a^{2} - \frac{620734424706820100496695143310368641556577965269327895619025545}{1628644715959703416657155680631139462235511401764794075210142842} a + \frac{587935410483793640381958086437965379714403914925727943601843345}{6514578863838813666628622722524557848942045607059176300840571368}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $14$
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:  \( 808862563918000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T362:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 2160
The 33 conjugacy class representatives for t18n362
Character table for t18n362 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.4.3196661779891.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.6.0.1}{6} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{4}$ $15{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R $15{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ $15{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ $15{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ $15{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
14731Data not computed