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Properties

Label 18.0.12904301529...0272.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{20}\cdot 3^{21}\cdot 7^{6}$
Root discriminant $14.89$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
Galois group $S_3^2$ (as 18T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12, 36, 48, 36, 12, 12, 40, 48, 36, 36, 54, 36, -11, -21, 0, 9, 0, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 9*x^15 - 21*x^13 - 11*x^12 + 36*x^11 + 54*x^10 + 36*x^9 + 36*x^8 + 48*x^7 + 40*x^6 + 12*x^5 + 12*x^4 + 36*x^3 + 48*x^2 + 36*x + 12)
 
gp: K = bnfinit(x^18 - 3*x^17 + 9*x^15 - 21*x^13 - 11*x^12 + 36*x^11 + 54*x^10 + 36*x^9 + 36*x^8 + 48*x^7 + 40*x^6 + 12*x^5 + 12*x^4 + 36*x^3 + 48*x^2 + 36*x + 12, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 9 x^{15} - 21 x^{13} - 11 x^{12} + 36 x^{11} + 54 x^{10} + 36 x^{9} + 36 x^{8} + 48 x^{7} + 40 x^{6} + 12 x^{5} + 12 x^{4} + 36 x^{3} + 48 x^{2} + 36 x + 12 \)

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:  \(-1290430152944907190272=-\,2^{20}\cdot 3^{21}\cdot 7^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.89$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{13} + \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} + \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{3196} a^{16} - \frac{147}{1598} a^{15} - \frac{201}{3196} a^{14} + \frac{497}{3196} a^{13} + \frac{719}{3196} a^{12} - \frac{1487}{3196} a^{11} + \frac{358}{799} a^{10} + \frac{883}{3196} a^{9} + \frac{381}{799} a^{8} - \frac{219}{1598} a^{7} - \frac{67}{1598} a^{6} + \frac{195}{1598} a^{5} - \frac{10}{799} a^{4} + \frac{279}{1598} a^{3} - \frac{19}{799} a^{2} + \frac{150}{799} a - \frac{308}{799}$, $\frac{1}{8292542948} a^{17} - \frac{1290815}{8292542948} a^{16} + \frac{220867403}{8292542948} a^{15} + \frac{17957492}{2073135737} a^{14} - \frac{402092894}{2073135737} a^{13} - \frac{207732658}{2073135737} a^{12} + \frac{1680662861}{8292542948} a^{11} - \frac{3401600789}{8292542948} a^{10} - \frac{1522962227}{8292542948} a^{9} - \frac{683230883}{2073135737} a^{8} + \frac{646465384}{2073135737} a^{7} - \frac{74034023}{4146271474} a^{6} - \frac{466154351}{4146271474} a^{5} + \frac{1105779049}{4146271474} a^{4} - \frac{849086543}{4146271474} a^{3} + \frac{822052286}{2073135737} a^{2} - \frac{202339493}{2073135737} a - \frac{41115909}{121949161}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{498523}{2594663} a^{17} - \frac{7355775}{10378652} a^{16} + \frac{5073111}{10378652} a^{15} + \frac{14032737}{10378652} a^{14} - \frac{3937583}{5189326} a^{13} - \frac{19117281}{5189326} a^{12} + \frac{232203}{5189326} a^{11} + \frac{79294497}{10378652} a^{10} + \frac{56902111}{10378652} a^{9} + \frac{22561623}{10378652} a^{8} + \frac{11548423}{2594663} a^{7} + \frac{33505185}{5189326} a^{6} + \frac{17986831}{5189326} a^{5} - \frac{2955675}{5189326} a^{4} + \frac{16895485}{5189326} a^{3} + \frac{28293117}{5189326} a^{2} + \frac{14093592}{2594663} a + \frac{8535064}{2594663} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 11704.438801595428 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T11):

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

\(\Q(\sqrt{-3}) \), 3.1.756.1, 3.1.108.1 x3, 6.0.1714608.3, 6.0.34992.1, 9.1.20739898368.1 x3

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$2$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.12.16.13$x^{12} + 12 x^{10} + 12 x^{8} + 8 x^{6} + 32 x^{4} - 16 x^{2} + 16$$6$$2$$16$$D_6$$[2]_{3}^{2}$
$3$3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
$7$7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$