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Properties

Label 18.0.19681004463...9616.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 31^{9}\cdot 127^{14}$
Root discriminant $481.95$
Ramified primes $2, 31, 127$
Class number $19765766286$ (GRH)
Class group $[9, 9, 244021806]$ (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![59975185408, -50298266368, 32946159232, -12494373536, 4235753024, -1127712400, 352111696, -92692658, 21773822, -3007341, 478943, -134342, 51598, -8092, 128, 184, 4, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 4*x^16 + 184*x^15 + 128*x^14 - 8092*x^13 + 51598*x^12 - 134342*x^11 + 478943*x^10 - 3007341*x^9 + 21773822*x^8 - 92692658*x^7 + 352111696*x^6 - 1127712400*x^5 + 4235753024*x^4 - 12494373536*x^3 + 32946159232*x^2 - 50298266368*x + 59975185408)
 
gp: K = bnfinit(x^18 - 7*x^17 + 4*x^16 + 184*x^15 + 128*x^14 - 8092*x^13 + 51598*x^12 - 134342*x^11 + 478943*x^10 - 3007341*x^9 + 21773822*x^8 - 92692658*x^7 + 352111696*x^6 - 1127712400*x^5 + 4235753024*x^4 - 12494373536*x^3 + 32946159232*x^2 - 50298266368*x + 59975185408, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 4 x^{16} + 184 x^{15} + 128 x^{14} - 8092 x^{13} + 51598 x^{12} - 134342 x^{11} + 478943 x^{10} - 3007341 x^{9} + 21773822 x^{8} - 92692658 x^{7} + 352111696 x^{6} - 1127712400 x^{5} + 4235753024 x^{4} - 12494373536 x^{3} + 32946159232 x^{2} - 50298266368 x + 59975185408 \)

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:  \(-1968100446373322200925262150357418865721092079616=-\,2^{18}\cdot 31^{9}\cdot 127^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $481.95$
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:  $2, 31, 127$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} - \frac{1}{16} a^{6} + \frac{1}{16} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} + \frac{1}{16} a^{7} + \frac{1}{16} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{64} a^{12} - \frac{1}{32} a^{11} - \frac{1}{64} a^{10} + \frac{1}{64} a^{8} + \frac{3}{32} a^{7} + \frac{5}{64} a^{6} + \frac{3}{16} a^{5} - \frac{7}{32} a^{4} - \frac{1}{4} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{13} - \frac{1}{64} a^{11} - \frac{1}{32} a^{10} - \frac{3}{64} a^{9} + \frac{5}{64} a^{7} + \frac{3}{32} a^{6} + \frac{7}{32} a^{5} - \frac{1}{16} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{128} a^{14} - \frac{1}{128} a^{13} - \frac{1}{128} a^{12} + \frac{3}{128} a^{11} - \frac{1}{128} a^{10} - \frac{1}{128} a^{9} - \frac{3}{128} a^{8} + \frac{5}{128} a^{7} + \frac{1}{16} a^{6} - \frac{15}{64} a^{5} - \frac{5}{32} a^{4} + \frac{3}{16} a^{3} - \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{128} a^{15} - \frac{1}{32} a^{11} - \frac{1}{32} a^{10} - \frac{1}{64} a^{9} + \frac{3}{128} a^{7} + \frac{3}{32} a^{6} - \frac{11}{64} a^{5} - \frac{1}{16} a^{4} + \frac{3}{16} a^{3}$, $\frac{1}{512} a^{16} - \frac{1}{256} a^{14} + \frac{1}{256} a^{13} + \frac{1}{256} a^{12} + \frac{7}{256} a^{11} + \frac{3}{128} a^{10} - \frac{15}{256} a^{9} + \frac{13}{512} a^{8} + \frac{13}{256} a^{7} + \frac{31}{256} a^{6} - \frac{23}{128} a^{5} + \frac{5}{64} a^{4} + \frac{13}{32} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{10018602273575239020506163584658702078079351830500031056384} a^{17} - \frac{8415780650706130279148215425010844032920875115674943}{5009301136787619510253081792329351039039675915250015528192} a^{16} - \frac{8668091392662279978534463268835578216323363930090862793}{5009301136787619510253081792329351039039675915250015528192} a^{15} + \frac{9520777010695314679520676661442449329224942719011263199}{5009301136787619510253081792329351039039675915250015528192} a^{14} + \frac{9601056767527960465299274559260255600594338672435092903}{5009301136787619510253081792329351039039675915250015528192} a^{13} + \frac{27031586580695119564117679676333849319523937863540098061}{5009301136787619510253081792329351039039675915250015528192} a^{12} + \frac{17795199045961290656062127858528984791316974357912065167}{626162642098452438781635224041168879879959489406251941024} a^{11} - \frac{113005577185980003129557855660324524357367131982018826015}{5009301136787619510253081792329351039039675915250015528192} a^{10} + \frac{371502861970221797657029817009425479549174009135994488665}{10018602273575239020506163584658702078079351830500031056384} a^{9} - \frac{29013772428283686150938384258009117947854752374363731361}{2504650568393809755126540896164675519519837957625007764096} a^{8} + \frac{371704997744547580128813788180938518425892777314963495869}{5009301136787619510253081792329351039039675915250015528192} a^{7} - \frac{50554018683869476086759957565778317912633691014058575573}{1252325284196904877563270448082337759759918978812503882048} a^{6} - \frac{66620952953361422248593990267753965282037411018472102349}{313081321049226219390817612020584439939979744703125970512} a^{5} + \frac{120066024908882200629367667587521642213743769018892566981}{626162642098452438781635224041168879879959489406251941024} a^{4} + \frac{13900360136726128071849573891711524959353004448086477145}{313081321049226219390817612020584439939979744703125970512} a^{3} + \frac{77441829863282004690883420908270198926661690645669046963}{156540660524613109695408806010292219969989872351562985256} a^{2} + \frac{9339553680997057919956326226855227286239705517286884993}{19567582565576638711926100751286527496248734043945373157} a + \frac{2031206124551919727657661309472095534193080201077365238}{19567582565576638711926100751286527496248734043945373157}$

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

Class group and class number

$C_{9}\times C_{9}\times C_{244021806}$, which has order $19765766286$ (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:  \( 5546046730.2947445 \) (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

\(\Q(\sqrt{-31}) \), 3.3.16129.1, 3.3.1016.1, 6.0.7749969000031.1, 6.0.30751938496.4, 9.9.272832440404737536.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
$31$31.6.3.2$x^{6} - 961 x^{2} + 268119$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31.12.6.1$x^{12} + 178746 x^{6} - 114516604 x^{2} + 7987533129$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$127$127.6.4.1$x^{6} + 1016 x^{3} + 435483$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
127.12.10.1$x^{12} - 1270 x^{6} + 11758041$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$