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Properties

Label 18.2.33966586417...8912.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{18}\cdot 3^{9}\cdot 37^{12}$
Root discriminant $38.46$
Ramified primes $2, 3, 37$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group $S_3^2$ (as 18T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-972, 0, -1692, 0, -2259, 0, 301, 0, 468, 0, 150, 0, -166, 0, 36, 0, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^16 + 36*x^14 - 166*x^12 + 150*x^10 + 468*x^8 + 301*x^6 - 2259*x^4 - 1692*x^2 - 972)
 
gp: K = bnfinit(x^18 - 3*x^16 + 36*x^14 - 166*x^12 + 150*x^10 + 468*x^8 + 301*x^6 - 2259*x^4 - 1692*x^2 - 972, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{16} + 36 x^{14} - 166 x^{12} + 150 x^{10} + 468 x^{8} + 301 x^{6} - 2259 x^{4} - 1692 x^{2} - 972 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(33966586417892403297890598912=2^{18}\cdot 3^{9}\cdot 37^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.46$
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:  $2, 3, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{12} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{12} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{12} a^{9} - \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} + \frac{1}{12} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{10} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{5} - \frac{1}{4} a^{4} - \frac{1}{6} a^{3} - \frac{1}{6} a^{2}$, $\frac{1}{12} a^{11} - \frac{1}{6} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{6} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{36} a^{12} + \frac{1}{36} a^{10} - \frac{1}{6} a^{7} - \frac{5}{36} a^{6} + \frac{1}{6} a^{5} + \frac{7}{36} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{72} a^{13} + \frac{1}{72} a^{11} - \frac{1}{24} a^{9} + \frac{7}{72} a^{7} + \frac{13}{72} a^{5} - \frac{5}{24} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{288} a^{14} + \frac{1}{96} a^{12} + \frac{5}{288} a^{10} - \frac{11}{288} a^{8} - \frac{1}{6} a^{7} + \frac{5}{96} a^{6} + \frac{1}{6} a^{5} - \frac{79}{288} a^{4} - \frac{1}{6} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2} a + \frac{1}{8}$, $\frac{1}{288} a^{15} - \frac{1}{288} a^{13} + \frac{1}{288} a^{11} + \frac{1}{288} a^{9} - \frac{13}{288} a^{7} - \frac{1}{6} a^{6} - \frac{131}{288} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{6} a^{2} - \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{872672256} a^{16} + \frac{155101}{109084032} a^{14} - \frac{953081}{218168064} a^{12} + \frac{9565703}{436336128} a^{10} + \frac{1768919}{54542016} a^{8} - \frac{33521495}{218168064} a^{6} - \frac{1}{2} a^{5} - \frac{106866797}{290890752} a^{4} - \frac{1}{2} a^{3} - \frac{435613}{1136292} a^{2} - \frac{10848623}{24240896}$, $\frac{1}{2618016768} a^{17} + \frac{177955}{109084032} a^{15} - \frac{570203}{218168064} a^{13} + \frac{11080759}{1309008384} a^{11} + \frac{2167823}{54542016} a^{9} - \frac{14456453}{218168064} a^{7} - \frac{1}{6} a^{6} + \frac{591463321}{2618016768} a^{5} - \frac{1}{3} a^{4} + \frac{195689}{568146} a^{3} + \frac{1}{3} a^{2} - \frac{1758287}{72722688} a - \frac{1}{2}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  $9$
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:  \( 8602131.533624997 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{3}) \), 3.1.5476.1, 3.1.4107.1, 6.2.2365632.1 x2, 6.2.3238550208.12, 6.2.3238550208.6, 9.1.4433575234752.1

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 6 sibling: data not computed
Degree 9 sibling: data not computed
Degree 12 sibling: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$37$37.9.6.1$x^{9} + 222 x^{6} + 15059 x^{3} + 405224$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
37.9.6.1$x^{9} + 222 x^{6} + 15059 x^{3} + 405224$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$