↑

Properties

Label 18.0.13890217521...5227.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 8677^{12}$
Root discriminant $2194.13$
Ramified primes $3, 8677$
Class number $54615837276$ (GRH)
Class group $[3, 3, 3, 9, 9, 1386, 18018]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![195920474112, 0, 0, 0, 0, 0, 46823693841, 0, 0, 0, 0, 0, 485982, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 485982*x^12 + 46823693841*x^6 + 195920474112)
 
gp: K = bnfinit(x^18 + 485982*x^12 + 46823693841*x^6 + 195920474112, 1)
 

Normalized defining polynomial

\( x^{18} + 485982 x^{12} + 46823693841 x^{6} + 195920474112 \)

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:  \(-1389021752112961964809440433226376447489723810648175849165227=-\,3^{27}\cdot 8677^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2194.13$
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:  $3, 8677$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{18} a^{6} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{396} a^{7} - \frac{1}{6} a^{5} - \frac{1}{2} a^{2} + \frac{37}{132} a$, $\frac{1}{792} a^{8} - \frac{95}{264} a^{2}$, $\frac{1}{792} a^{9} - \frac{95}{264} a^{3}$, $\frac{1}{792} a^{10} + \frac{37}{264} a^{4} - \frac{1}{2} a$, $\frac{1}{4752} a^{11} - \frac{1}{1584} a^{8} - \frac{359}{1584} a^{5} - \frac{1}{3} a^{3} - \frac{169}{528} a^{2}$, $\frac{1}{265122182160} a^{12} - \frac{1}{2376} a^{10} - \frac{1}{1584} a^{9} - \frac{194443499}{17674812144} a^{6} + \frac{95}{792} a^{4} - \frac{169}{528} a^{3} + \frac{1}{3} a^{2} - \frac{78391066}{167375115}$, $\frac{1}{265122182160} a^{13} - \frac{1}{1584} a^{10} - \frac{15910043}{17674812144} a^{7} + \frac{95}{528} a^{4} - \frac{1}{3} a^{3} + \frac{562856453}{3682252530} a$, $\frac{1}{2916344003760} a^{14} - \frac{24692533}{97211466792} a^{8} - \frac{1}{6} a^{4} + \frac{20403487549}{324038222640} a^{2} - \frac{1}{2} a$, $\frac{1}{256638272330880} a^{15} - \frac{2725011055}{8554609077696} a^{9} - \frac{1}{6} a^{5} + \frac{9390508758889}{28515363592320} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{33876251947676160} a^{16} + \frac{105287729825}{1129208398255872} a^{10} + \frac{1}{2376} a^{8} - \frac{1}{792} a^{7} - \frac{1}{6} a^{5} - \frac{877394093865911}{3764027994186240} a^{4} + \frac{37}{792} a^{2} - \frac{37}{264} a - \frac{1}{3}$, $\frac{1}{1490555085697751040} a^{17} - \frac{1}{5832688007520} a^{14} - \frac{3696760749151}{49685169523258368} a^{11} - \frac{1}{2376} a^{9} - \frac{49024609}{97211466792} a^{8} + \frac{14653973942751049}{165617231744194560} a^{5} - \frac{1}{6} a^{4} + \frac{95}{792} a^{3} + \frac{96201175901}{648076445280} a^{2} - \frac{1}{6} a$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{9}\times C_{9}\times C_{1386}\times C_{18018}$, which has order $54615837276$ (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:  \( \frac{1}{111824955264} a^{15} + \frac{243959}{55912477632} a^{9} + \frac{15781574571}{37274985088} a^{3} + \frac{1}{2} \) (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:  \( 98053733517680.33 \) (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

\(\Q(\sqrt{-3}) \), Deg 3 x3, Deg 3, 6.0.153053108305062507.1, 6.0.111575715954390567603.1, Deg 6 x2, 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$3$3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
8677Data not computed