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Properties

Label 18.0.49066952384...1488.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 7^{12}\cdot 2953^{12}$
Root discriminant $2070.88$
Ramified primes $2, 3, 7, 2953$
Class number $28298170368$ (GRH)
Class group $[2, 6, 6, 18, 36, 36, 36, 468]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![832184947467, 0, -170682814251, 0, 5576982228, 0, 2448021928, 0, -64460646, 0, -268014, 0, 195268, 0, 1008, 0, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 3*x^16 + 1008*x^14 + 195268*x^12 - 268014*x^10 - 64460646*x^8 + 2448021928*x^6 + 5576982228*x^4 - 170682814251*x^2 + 832184947467)
 
gp: K = bnfinit(x^18 + 3*x^16 + 1008*x^14 + 195268*x^12 - 268014*x^10 - 64460646*x^8 + 2448021928*x^6 + 5576982228*x^4 - 170682814251*x^2 + 832184947467, 1)
 

Normalized defining polynomial

\( x^{18} + 3 x^{16} + 1008 x^{14} + 195268 x^{12} - 268014 x^{10} - 64460646 x^{8} + 2448021928 x^{6} + 5576982228 x^{4} - 170682814251 x^{2} + 832184947467 \)

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:  \(-490669523847518219543603050415476754595311155180270121791488=-\,2^{12}\cdot 3^{9}\cdot 7^{12}\cdot 2953^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2070.88$
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:  $2, 3, 7, 2953$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{12} a^{6} + \frac{1}{12} a^{4} - \frac{5}{12} a^{2} + \frac{1}{4}$, $\frac{1}{12} a^{7} + \frac{1}{12} a^{5} + \frac{1}{12} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{324} a^{8} + \frac{7}{54} a^{4} + \frac{26}{81} a^{2} - \frac{49}{108}$, $\frac{1}{648} a^{9} - \frac{1}{648} a^{8} + \frac{7}{108} a^{5} - \frac{7}{108} a^{4} + \frac{13}{81} a^{3} - \frac{13}{81} a^{2} - \frac{49}{216} a + \frac{49}{216}$, $\frac{1}{1296} a^{10} - \frac{1}{1296} a^{8} - \frac{1}{24} a^{7} - \frac{1}{108} a^{6} - \frac{1}{24} a^{5} + \frac{1}{162} a^{4} + \frac{5}{24} a^{3} + \frac{19}{1296} a^{2} + \frac{3}{8} a + \frac{211}{432}$, $\frac{1}{1296} a^{11} - \frac{1}{1296} a^{9} - \frac{1}{648} a^{8} - \frac{1}{108} a^{7} - \frac{1}{24} a^{6} + \frac{1}{162} a^{5} - \frac{23}{216} a^{4} + \frac{19}{1296} a^{3} - \frac{293}{648} a^{2} + \frac{211}{432} a + \frac{11}{108}$, $\frac{1}{104976} a^{12} + \frac{1}{52488} a^{10} + \frac{13}{34992} a^{8} + \frac{113}{6561} a^{6} - \frac{8165}{104976} a^{4} - \frac{4691}{17496} a^{2} + \frac{3829}{11664}$, $\frac{1}{209952} a^{13} - \frac{1}{209952} a^{12} + \frac{1}{104976} a^{11} - \frac{1}{104976} a^{10} + \frac{13}{69984} a^{9} - \frac{13}{69984} a^{8} + \frac{113}{13122} a^{7} - \frac{113}{13122} a^{6} + \frac{44323}{209952} a^{5} + \frac{8165}{209952} a^{4} - \frac{4691}{34992} a^{3} - \frac{12805}{34992} a^{2} + \frac{9661}{23328} a - \frac{3829}{23328}$, $\frac{1}{1422844704} a^{14} + \frac{3349}{1422844704} a^{12} - \frac{398753}{1422844704} a^{10} + \frac{1780043}{1422844704} a^{8} - \frac{1}{24} a^{7} - \frac{26602381}{1422844704} a^{6} + \frac{5}{24} a^{5} - \frac{14836393}{1422844704} a^{4} + \frac{5}{24} a^{3} - \frac{124155857}{474281568} a^{2} + \frac{1}{8} a + \frac{12138449}{158093856}$, $\frac{1}{1422844704} a^{15} + \frac{3349}{1422844704} a^{13} - \frac{398753}{1422844704} a^{11} - \frac{415705}{1422844704} a^{9} - \frac{26602381}{1422844704} a^{7} - \frac{1}{24} a^{6} - \frac{107057809}{1422844704} a^{5} + \frac{5}{24} a^{4} + \frac{36865663}{474281568} a^{3} + \frac{5}{24} a^{2} - \frac{31044595}{158093856} a + \frac{1}{8}$, $\frac{1}{79056478003233089117376} a^{16} + \frac{4522982816945}{39528239001616544558688} a^{14} - \frac{164622050608597909}{39528239001616544558688} a^{12} - \frac{9404152094768965475}{39528239001616544558688} a^{10} + \frac{4971145766831051609}{4941029875202068069836} a^{8} - \frac{1413784440079765272893}{39528239001616544558688} a^{6} - \frac{1}{4} a^{5} + \frac{481554815820206281525}{4392026555735171617632} a^{4} - \frac{638573188316483614051}{1464008851911723872544} a^{2} - \frac{1}{4} a - \frac{852146467179022700563}{2928017703823447745088}$, $\frac{1}{13879234334725604358535647936} a^{17} + \frac{581300518236396251}{1734904291840700544816955992} a^{15} - \frac{12540697848734675270929}{6939617167362802179267823968} a^{13} - \frac{1}{209952} a^{12} - \frac{1205998687146544146061873}{3469808583681401089633911984} a^{11} - \frac{1}{104976} a^{10} + \frac{2235911762192428179089585}{3469808583681401089633911984} a^{9} - \frac{13}{69984} a^{8} - \frac{47009722066836438902665109}{1734904291840700544816955992} a^{7} - \frac{113}{13122} a^{6} + \frac{120222728708500055710451881}{771068574151422464363091552} a^{5} + \frac{8165}{209952} a^{4} - \frac{7108284197051614392771137}{128511429025237077393848592} a^{3} - \frac{12805}{34992} a^{2} - \frac{47151058762687664818896223}{514045716100948309575394368} a + \frac{7835}{23328}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{6}\times C_{18}\times C_{36}\times C_{36}\times C_{36}\times C_{468}$, which has order $28298170368$ (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:  \( \frac{516304348573}{220301810046278699679936} a^{17} + \frac{720064885457}{27537726255784837459992} a^{15} + \frac{8478033358066}{3442215781973104682499} a^{13} + \frac{3286168276318543}{6884431563946209364998} a^{11} + \frac{353236481261555725}{110150905023139349839968} a^{9} - \frac{4066193272590172397}{27537726255784837459992} a^{7} + \frac{14217047269124766715}{3059747361753870828888} a^{5} + \frac{8267299427135731393}{127489473406411284537} a^{3} - \frac{1979636464787085161917}{8159326298010322210368} a + \frac{1}{2} \) (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:  \( 63617903134123.17 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.5127482892.1 x3, 3.3.427290241.1, 6.0.78873242423258050992.1, 6.0.184589384112.1 x2, 6.0.4929577651453628187.1, Deg 9 x3

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

Sibling fields

Degree 6 sibling: data not computed
Degree 9 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
2Data not computed
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
2953Data not computed