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Properties

Label 18.0.81916056571...5043.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{21}\cdot 23^{8}$
Root discriminant $14.52$
Ramified primes $3, 23$
Class number $1$
Class group Trivial
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![1, -12, 69, -247, 606, -1023, 1138, -738, 78, 379, -390, 123, 100, -150, 99, -45, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 18*x^16 - 45*x^15 + 99*x^14 - 150*x^13 + 100*x^12 + 123*x^11 - 390*x^10 + 379*x^9 + 78*x^8 - 738*x^7 + 1138*x^6 - 1023*x^5 + 606*x^4 - 247*x^3 + 69*x^2 - 12*x + 1)
 
gp: K = bnfinit(x^18 - 6*x^17 + 18*x^16 - 45*x^15 + 99*x^14 - 150*x^13 + 100*x^12 + 123*x^11 - 390*x^10 + 379*x^9 + 78*x^8 - 738*x^7 + 1138*x^6 - 1023*x^5 + 606*x^4 - 247*x^3 + 69*x^2 - 12*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 18 x^{16} - 45 x^{15} + 99 x^{14} - 150 x^{13} + 100 x^{12} + 123 x^{11} - 390 x^{10} + 379 x^{9} + 78 x^{8} - 738 x^{7} + 1138 x^{6} - 1023 x^{5} + 606 x^{4} - 247 x^{3} + 69 x^{2} - 12 x + 1 \)

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:  \(-819160565714194205043=-\,3^{21}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.52$
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:  $3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{14} a^{15} - \frac{3}{14} a^{14} + \frac{3}{14} a^{11} - \frac{1}{14} a^{10} - \frac{1}{2} a^{9} - \frac{3}{14} a^{8} + \frac{3}{14} a^{7} - \frac{5}{14} a^{6} - \frac{2}{7} a^{5} - \frac{1}{2} a^{4} + \frac{3}{7} a^{3} - \frac{3}{7} a^{2} + \frac{1}{7} a + \frac{5}{14}$, $\frac{1}{14} a^{16} - \frac{1}{7} a^{14} + \frac{3}{14} a^{12} - \frac{3}{7} a^{11} - \frac{3}{14} a^{10} + \frac{2}{7} a^{9} + \frac{1}{14} a^{8} + \frac{2}{7} a^{7} + \frac{1}{7} a^{6} - \frac{5}{14} a^{5} - \frac{1}{14} a^{4} + \frac{5}{14} a^{3} + \frac{5}{14} a^{2} + \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{3587142986} a^{17} - \frac{50184383}{1793571493} a^{16} - \frac{15176239}{512448998} a^{15} + \frac{49087498}{1793571493} a^{14} - \frac{221178330}{1793571493} a^{13} - \frac{155760368}{1793571493} a^{12} + \frac{1204764465}{3587142986} a^{11} + \frac{422092768}{1793571493} a^{10} + \frac{633927121}{1793571493} a^{9} + \frac{117640812}{1793571493} a^{8} + \frac{230864930}{1793571493} a^{7} + \frac{656769439}{3587142986} a^{6} + \frac{677475991}{3587142986} a^{5} - \frac{800951946}{1793571493} a^{4} + \frac{752434717}{1793571493} a^{3} - \frac{245269986}{1793571493} a^{2} - \frac{173503718}{1793571493} a + \frac{380696387}{1793571493}$

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:  \( -\frac{51979339}{3429391} a^{17} + \frac{279634776}{3429391} a^{16} - \frac{763174681}{3429391} a^{15} + \frac{1870676501}{3429391} a^{14} - \frac{3998352901}{3429391} a^{13} + \frac{5347867137}{3429391} a^{12} - \frac{1945799574}{3429391} a^{11} - \frac{7522234641}{3429391} a^{10} + \frac{15597623991}{3429391} a^{9} - \frac{10170608314}{3429391} a^{8} - \frac{10121991233}{3429391} a^{7} + \frac{31978735966}{3429391} a^{6} - \frac{39542799466}{3429391} a^{5} + \frac{29164636624}{3429391} a^{4} - \frac{13969705025}{3429391} a^{3} + \frac{4532439157}{3429391} a^{2} - \frac{925787454}{3429391} a + \frac{94308798}{3429391} \) (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:  \( 2958.28701084 \)
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

\(\Q(\sqrt{-3}) \), 3.3.621.1, 3.1.23.1, 6.0.1156923.2 x2, 6.0.1156923.1, 6.0.14283.1, 9.3.5508110403.1

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: 6.0.1156923.2
Degree 9 sibling: 9.3.5508110403.1
Degree 12 sibling: 12.0.708051067974441.1
Degree 18 siblings: 18.0.697803444867646915407.1, 18.6.18840693011426466715989.1

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
3Data not computed
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$