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Properties

Label 18.0.31238944675...0016.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{33}\cdot 7^{9}\cdot 37^{14}$
Root discriminant $156.37$
Ramified primes $2, 7, 37$
Class number $3015376$ (GRH)
Class group $[2, 2, 753844]$ (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![97158937289, 8074688936, 41322035770, 1747923682, 7988920261, 68658878, 940014547, -16921424, 75342043, -2774640, 4318608, -203038, 178994, -8562, 5245, -202, 101, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 101*x^16 - 202*x^15 + 5245*x^14 - 8562*x^13 + 178994*x^12 - 203038*x^11 + 4318608*x^10 - 2774640*x^9 + 75342043*x^8 - 16921424*x^7 + 940014547*x^6 + 68658878*x^5 + 7988920261*x^4 + 1747923682*x^3 + 41322035770*x^2 + 8074688936*x + 97158937289)
 
gp: K = bnfinit(x^18 - 2*x^17 + 101*x^16 - 202*x^15 + 5245*x^14 - 8562*x^13 + 178994*x^12 - 203038*x^11 + 4318608*x^10 - 2774640*x^9 + 75342043*x^8 - 16921424*x^7 + 940014547*x^6 + 68658878*x^5 + 7988920261*x^4 + 1747923682*x^3 + 41322035770*x^2 + 8074688936*x + 97158937289, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 101 x^{16} - 202 x^{15} + 5245 x^{14} - 8562 x^{13} + 178994 x^{12} - 203038 x^{11} + 4318608 x^{10} - 2774640 x^{9} + 75342043 x^{8} - 16921424 x^{7} + 940014547 x^{6} + 68658878 x^{5} + 7988920261 x^{4} + 1747923682 x^{3} + 41322035770 x^{2} + 8074688936 x + 97158937289 \)

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:  \(-3123894467595344662196894875152879190016=-\,2^{33}\cdot 7^{9}\cdot 37^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $156.37$
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:  $2, 7, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{10} + \frac{1}{6} a^{9} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{1}{3} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6} a$, $\frac{1}{6} a^{14} + \frac{1}{6} a^{11} - \frac{1}{6} a^{10} + \frac{1}{6} a^{9} - \frac{1}{3} a^{8} - \frac{1}{2} a^{7} + \frac{1}{6} a^{6} + \frac{1}{3} a^{5} - \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{11} - \frac{1}{6} a^{10} + \frac{1}{6} a^{7} - \frac{1}{3} a^{6} - \frac{1}{6} a^{5} + \frac{1}{3} a^{4} + \frac{1}{6} a^{3} + \frac{1}{6} a^{2} + \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{822} a^{16} - \frac{8}{411} a^{15} - \frac{14}{411} a^{14} - \frac{53}{822} a^{13} - \frac{7}{274} a^{12} + \frac{83}{411} a^{11} + \frac{46}{411} a^{10} - \frac{7}{274} a^{9} - \frac{5}{137} a^{8} - \frac{46}{411} a^{7} - \frac{163}{411} a^{6} - \frac{115}{411} a^{5} - \frac{337}{822} a^{4} + \frac{52}{411} a^{3} + \frac{19}{411} a^{2} + \frac{23}{822} a + \frac{191}{411}$, $\frac{1}{36425882306863751822295885167007242880001786184144761347822} a^{17} - \frac{2156808829541466996618700028805629699745226094382020201}{12141960768954583940765295055669080960000595394714920449274} a^{16} - \frac{2034073426305593055810833123903885843779291511389086485131}{36425882306863751822295885167007242880001786184144761347822} a^{15} + \frac{2835681531781215131482649198818868565315880797208516890779}{36425882306863751822295885167007242880001786184144761347822} a^{14} + \frac{2372723622990791984059173636728486242928173818105637357385}{36425882306863751822295885167007242880001786184144761347822} a^{13} + \frac{409364973294307090702823864648298045173306174538111195603}{12141960768954583940765295055669080960000595394714920449274} a^{12} - \frac{2271526135861227392202596087202416269919022000753291013383}{12141960768954583940765295055669080960000595394714920449274} a^{11} + \frac{7945623116599468806276266393578962145623537556555251311527}{36425882306863751822295885167007242880001786184144761347822} a^{10} + \frac{1824683320435926792010360639690091344737635080780621824683}{18212941153431875911147942583503621440000893092072380673911} a^{9} - \frac{2602312793833743148776547132376989143747260773534141732160}{6070980384477291970382647527834540480000297697357460224637} a^{8} - \frac{7022278530830240997841858439613393351145278590551546039549}{18212941153431875911147942583503621440000893092072380673911} a^{7} + \frac{6312824295280107893897648726298789660238187358975444535598}{18212941153431875911147942583503621440000893092072380673911} a^{6} + \frac{1484003424153308305774775458064054927924664697840149430453}{12141960768954583940765295055669080960000595394714920449274} a^{5} - \frac{1244426106370898728273369645370648348862953510245683350643}{12141960768954583940765295055669080960000595394714920449274} a^{4} + \frac{1357755938799239008207177247674174802213758756527275751765}{12141960768954583940765295055669080960000595394714920449274} a^{3} - \frac{4939653020845449378571029726549508840326315240221700461423}{12141960768954583940765295055669080960000595394714920449274} a^{2} - \frac{1729142332512028527932293492034807534810795182955029291988}{18212941153431875911147942583503621440000893092072380673911} a - \frac{34206986649182398915527935099327942099994946504620901715}{132941176302422451906189361923384098102196299942134165503}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{753844}$, which has order $3015376$ (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:  \( 615797.1340659427 \) (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

\(\Q(\sqrt{-14}) \), 3.3.1369.1, 3.3.148.1, 6.0.329132658176.5, 6.0.961673216.5, 9.9.6075640136512.1

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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$7$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$37$37.6.4.1$x^{6} + 518 x^{3} + 171125$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
37.12.10.1$x^{12} + 1998 x^{6} + 21390625$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$