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Properties

Label 18.0.54232587489...5472.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{31}\cdot 11^{8}$
Root discriminant $30.57$
Ramified primes $2, 3, 11$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3^2:S_3$ (as 18T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![196608, -147456, 516096, -304128, 594432, -293184, 393744, -165024, 163224, -57312, 43344, -12222, 7281, -1539, 744, -105, 42, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 42*x^16 - 105*x^15 + 744*x^14 - 1539*x^13 + 7281*x^12 - 12222*x^11 + 43344*x^10 - 57312*x^9 + 163224*x^8 - 165024*x^7 + 393744*x^6 - 293184*x^5 + 594432*x^4 - 304128*x^3 + 516096*x^2 - 147456*x + 196608)
 
gp: K = bnfinit(x^18 - 3*x^17 + 42*x^16 - 105*x^15 + 744*x^14 - 1539*x^13 + 7281*x^12 - 12222*x^11 + 43344*x^10 - 57312*x^9 + 163224*x^8 - 165024*x^7 + 393744*x^6 - 293184*x^5 + 594432*x^4 - 304128*x^3 + 516096*x^2 - 147456*x + 196608, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 42 x^{16} - 105 x^{15} + 744 x^{14} - 1539 x^{13} + 7281 x^{12} - 12222 x^{11} + 43344 x^{10} - 57312 x^{9} + 163224 x^{8} - 165024 x^{7} + 393744 x^{6} - 293184 x^{5} + 594432 x^{4} - 304128 x^{3} + 516096 x^{2} - 147456 x + 196608 \)

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:  \(-542325874896471806702825472=-\,2^{12}\cdot 3^{31}\cdot 11^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.57$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} + \frac{3}{8} a^{7} + \frac{1}{4} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} + \frac{3}{8} a^{6} - \frac{1}{2} a^{5} - \frac{1}{8} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{7}{16} a^{7} + \frac{3}{16} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{64} a^{13} + \frac{1}{64} a^{12} - \frac{1}{32} a^{11} - \frac{1}{64} a^{10} + \frac{1}{16} a^{9} - \frac{3}{64} a^{8} - \frac{27}{64} a^{7} + \frac{3}{32} a^{6} - \frac{1}{8} a^{5} - \frac{1}{2} a^{4} + \frac{3}{8} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a$, $\frac{1}{256} a^{14} + \frac{1}{256} a^{13} - \frac{1}{128} a^{12} - \frac{1}{256} a^{11} + \frac{1}{64} a^{10} + \frac{29}{256} a^{9} - \frac{59}{256} a^{8} + \frac{35}{128} a^{7} - \frac{13}{32} a^{6} - \frac{3}{8} a^{5} + \frac{15}{32} a^{4} - \frac{1}{2} a^{3} - \frac{7}{16} a^{2} - \frac{1}{2} a$, $\frac{1}{11264} a^{15} + \frac{13}{11264} a^{14} - \frac{35}{5632} a^{13} - \frac{201}{11264} a^{12} + \frac{3}{1408} a^{11} + \frac{189}{11264} a^{10} + \frac{1153}{11264} a^{9} + \frac{1289}{5632} a^{8} + \frac{255}{704} a^{7} - \frac{49}{352} a^{6} + \frac{57}{128} a^{5} + \frac{43}{352} a^{4} + \frac{17}{704} a^{3} - \frac{41}{176} a^{2} + \frac{1}{11} a + \frac{2}{11}$, $\frac{1}{585728} a^{16} - \frac{7}{585728} a^{15} - \frac{23}{26624} a^{14} - \frac{291}{53248} a^{13} + \frac{747}{146432} a^{12} + \frac{18189}{585728} a^{11} - \frac{20227}{585728} a^{10} + \frac{1141}{26624} a^{9} + \frac{15779}{73216} a^{8} + \frac{695}{4576} a^{7} + \frac{23203}{73216} a^{6} - \frac{931}{2288} a^{5} - \frac{13407}{36608} a^{4} - \frac{1295}{4576} a^{3} - \frac{21}{52} a^{2} + \frac{101}{286} a + \frac{12}{143}$, $\frac{1}{235926554886441582592} a^{17} + \frac{93439919015217}{235926554886441582592} a^{16} - \frac{5709411627093}{9074098264863137792} a^{15} + \frac{412688709049108335}{235926554886441582592} a^{14} + \frac{327071041531369757}{58981638721610395648} a^{13} + \frac{1921643007276744269}{235926554886441582592} a^{12} + \frac{453791954290827157}{235926554886441582592} a^{11} + \frac{5985030792310262115}{117963277443220791296} a^{10} - \frac{268774313749652467}{2268524566215784448} a^{9} + \frac{911936789913025963}{3686352420100649728} a^{8} + \frac{7740535119750645363}{29490819360805197824} a^{7} + \frac{84752943125599821}{283565570776973056} a^{6} - \frac{19846480546783841}{103114753009808384} a^{5} + \frac{84328746084748257}{460794052512581216} a^{4} + \frac{74379862959243519}{460794052512581216} a^{3} + \frac{2783674628525275}{17722848173560816} a^{2} + \frac{18498277191592597}{57599256564072652} a - \frac{3857180853361024}{14399814141018163}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( -\frac{120901323}{2291050749952} a^{17} + \frac{6179527125}{25201558249472} a^{16} - \frac{163160903}{71595335936} a^{15} + \frac{218397202557}{25201558249472} a^{14} - \frac{515165887209}{12600779124736} a^{13} + \frac{3249379832451}{25201558249472} a^{12} - \frac{10057785726663}{25201558249472} a^{11} + \frac{1654077749667}{1575097390592} a^{10} - \frac{14606007385625}{6300389562368} a^{9} + \frac{8096410524117}{1575097390592} a^{8} - \frac{25827317054859}{3150194781184} a^{7} + \frac{24809827806969}{1575097390592} a^{6} - \frac{27499076557737}{1575097390592} a^{5} + \frac{23497193206431}{787548695296} a^{4} - \frac{187178177781}{8949416992} a^{3} + \frac{98917212609}{3076362091} a^{2} - \frac{3256131231}{279669281} a + \frac{48010907768}{3076362091} \) (order $6$)
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:  \( 1643556.992454405 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 18T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.108.1 x3, 6.0.34992.1, 9.3.13445270232768.1

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

Sibling fields

Degree 9 siblings: data not computed
Degree 18 siblings: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
3Data not computed
$11$11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
11.6.4.2$x^{6} - 11 x^{3} + 847$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
11.6.4.2$x^{6} - 11 x^{3} + 847$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$