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Properties

Label 18.2.76936321186...1968.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{24}\cdot 3^{9}\cdot 13^{12}$
Root discriminant $24.13$
Ramified primes $2, 3, 13$
Class number $3$ (GRH)
Class group $[3]$ (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![1, 0, -1, -8, -18, -64, -106, -192, -292, -368, -292, -192, -106, -64, -18, -8, -1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^16 - 8*x^15 - 18*x^14 - 64*x^13 - 106*x^12 - 192*x^11 - 292*x^10 - 368*x^9 - 292*x^8 - 192*x^7 - 106*x^6 - 64*x^5 - 18*x^4 - 8*x^3 - x^2 + 1)
 
gp: K = bnfinit(x^18 - x^16 - 8*x^15 - 18*x^14 - 64*x^13 - 106*x^12 - 192*x^11 - 292*x^10 - 368*x^9 - 292*x^8 - 192*x^7 - 106*x^6 - 64*x^5 - 18*x^4 - 8*x^3 - x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{18} - x^{16} - 8 x^{15} - 18 x^{14} - 64 x^{13} - 106 x^{12} - 192 x^{11} - 292 x^{10} - 368 x^{9} - 292 x^{8} - 192 x^{7} - 106 x^{6} - 64 x^{5} - 18 x^{4} - 8 x^{3} - x^{2} + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7693632118668862546771968=2^{24}\cdot 3^{9}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.13$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{4} a^{5} + \frac{1}{4} a$, $\frac{1}{24} a^{14} - \frac{1}{12} a^{13} - \frac{1}{8} a^{12} + \frac{1}{12} a^{11} - \frac{1}{24} a^{10} - \frac{5}{24} a^{8} + \frac{1}{6} a^{7} - \frac{5}{24} a^{6} - \frac{1}{4} a^{5} + \frac{11}{24} a^{4} + \frac{1}{12} a^{3} + \frac{3}{8} a^{2} - \frac{1}{3} a + \frac{1}{24}$, $\frac{1}{48} a^{15} - \frac{1}{48} a^{14} + \frac{1}{48} a^{13} - \frac{1}{48} a^{12} - \frac{5}{48} a^{11} + \frac{5}{48} a^{10} + \frac{7}{48} a^{9} - \frac{7}{48} a^{8} + \frac{11}{48} a^{7} - \frac{11}{48} a^{6} + \frac{23}{48} a^{5} + \frac{1}{48} a^{4} + \frac{5}{48} a^{3} + \frac{19}{48} a^{2} + \frac{5}{48} a - \frac{5}{48}$, $\frac{1}{1584} a^{16} - \frac{7}{1584} a^{15} - \frac{19}{1584} a^{14} - \frac{1}{16} a^{13} + \frac{67}{1584} a^{12} - \frac{137}{1584} a^{11} + \frac{3}{176} a^{10} + \frac{383}{1584} a^{9} + \frac{89}{528} a^{8} - \frac{277}{1584} a^{7} + \frac{97}{528} a^{6} + \frac{787}{1584} a^{5} + \frac{133}{1584} a^{4} - \frac{23}{48} a^{3} - \frac{151}{1584} a^{2} + \frac{125}{1584} a + \frac{29}{198}$, $\frac{1}{3168} a^{17} - \frac{1}{3168} a^{16} - \frac{7}{792} a^{15} - \frac{1}{66} a^{14} + \frac{149}{1584} a^{13} + \frac{17}{1584} a^{12} - \frac{7}{132} a^{11} + \frac{29}{792} a^{10} + \frac{35}{264} a^{9} + \frac{13}{396} a^{8} + \frac{2}{11} a^{7} + \frac{97}{792} a^{6} - \frac{757}{1584} a^{5} - \frac{73}{176} a^{4} + \frac{251}{792} a^{3} + \frac{7}{18} a^{2} - \frac{1229}{3168} a - \frac{53}{1056}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 222009.1690420821 \) (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

\(\Q(\sqrt{3}) \), 3.1.676.1, 3.1.2028.1, 6.2.49353408.1, 6.2.1168128.1 x2, 6.2.197413632.1, 9.1.133451615232.2

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.6.8.3$x^{6} + 2 x^{3} + 6$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.12.16.13$x^{12} + 12 x^{10} + 12 x^{8} + 8 x^{6} + 32 x^{4} - 16 x^{2} + 16$$6$$2$$16$$D_6$$[2]_{3}^{2}$
$3$3.2.1.1$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$$[\ ]_{2}^{2}$
$13$13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$