↑

Properties

Label 18.0.26451111757...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 3^{6}\cdot 5^{6}\cdot 7^{12}$
Root discriminant $22.74$
Ramified primes $2, 3, 5, 7$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2\times S_3^2$ (as 18T29)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 2, 6, 10, -22, -51, 94, 42, -148, 42, 94, -51, -22, 10, 6, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 2*x^16 + 6*x^15 + 10*x^14 - 22*x^13 - 51*x^12 + 94*x^11 + 42*x^10 - 148*x^9 + 42*x^8 + 94*x^7 - 51*x^6 - 22*x^5 + 10*x^4 + 6*x^3 + 2*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^18 - 4*x^17 + 2*x^16 + 6*x^15 + 10*x^14 - 22*x^13 - 51*x^12 + 94*x^11 + 42*x^10 - 148*x^9 + 42*x^8 + 94*x^7 - 51*x^6 - 22*x^5 + 10*x^4 + 6*x^3 + 2*x^2 - 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 2 x^{16} + 6 x^{15} + 10 x^{14} - 22 x^{13} - 51 x^{12} + 94 x^{11} + 42 x^{10} - 148 x^{9} + 42 x^{8} + 94 x^{7} - 51 x^{6} - 22 x^{5} + 10 x^{4} + 6 x^{3} + 2 x^{2} - 4 x + 1 \)

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:  \(-2645111175781810176000000=-\,2^{24}\cdot 3^{6}\cdot 5^{6}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.74$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{11} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{1}{5} a^{10} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{110} a^{14} - \frac{4}{55} a^{12} - \frac{21}{110} a^{10} + \frac{19}{55} a^{9} + \frac{1}{10} a^{8} + \frac{24}{55} a^{7} - \frac{1}{10} a^{6} - \frac{5}{11} a^{5} - \frac{21}{110} a^{4} - \frac{1}{5} a^{3} - \frac{3}{11} a^{2} - \frac{1}{5} a - \frac{43}{110}$, $\frac{1}{110} a^{15} - \frac{4}{55} a^{13} - \frac{21}{110} a^{11} + \frac{19}{55} a^{10} + \frac{1}{10} a^{9} + \frac{24}{55} a^{8} - \frac{1}{10} a^{7} - \frac{5}{11} a^{6} - \frac{21}{110} a^{5} - \frac{1}{5} a^{4} - \frac{3}{11} a^{3} - \frac{1}{5} a^{2} - \frac{43}{110} a$, $\frac{1}{220} a^{16} - \frac{1}{220} a^{14} - \frac{1}{10} a^{13} - \frac{1}{20} a^{12} + \frac{4}{55} a^{11} - \frac{7}{22} a^{10} - \frac{41}{110} a^{9} + \frac{2}{5} a^{8} - \frac{1}{10} a^{7} - \frac{1}{22} a^{6} - \frac{16}{55} a^{5} - \frac{1}{220} a^{4} - \frac{1}{10} a^{3} - \frac{1}{20} a^{2} + \frac{51}{220}$, $\frac{1}{440} a^{17} - \frac{1}{440} a^{16} + \frac{1}{440} a^{15} + \frac{1}{440} a^{14} + \frac{39}{440} a^{13} + \frac{27}{440} a^{12} - \frac{1}{11} a^{11} - \frac{89}{220} a^{10} + \frac{37}{110} a^{9} - \frac{31}{110} a^{8} - \frac{27}{220} a^{7} - \frac{2}{5} a^{6} + \frac{109}{440} a^{5} - \frac{87}{440} a^{4} + \frac{127}{440} a^{3} + \frac{1}{8} a^{2} + \frac{97}{440} a - \frac{73}{440}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{35}{22} a^{17} + \frac{111}{20} a^{16} - \frac{13}{55} a^{15} - \frac{2237}{220} a^{14} - \frac{1132}{55} a^{13} + \frac{1093}{44} a^{12} + \frac{10421}{110} a^{11} - \frac{1043}{10} a^{10} - \frac{6814}{55} a^{9} + \frac{10169}{55} a^{8} + \frac{282}{11} a^{7} - \frac{16711}{110} a^{6} + \frac{818}{55} a^{5} + \frac{10523}{220} a^{4} + \frac{89}{55} a^{3} - \frac{2267}{220} a^{2} - \frac{383}{55} a + \frac{843}{220} \) (order $4$)
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:  \( 169260.62829015162 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_3^2$ (as 18T29):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 18 conjugacy class representatives for $C_2\times S_3^2$
Character table for $C_2\times S_3^2$

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.1.980.1, 3.1.588.1, 6.0.3841600.1, 6.0.5531904.1, 9.1.406594944000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 siblings: 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 R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.8.1$x^{6} + 2 x^{3} + 2$$6$$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$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7Data not computed