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Properties

Label 18.0.48085200741...3248.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{20}\cdot 3^{9}\cdot 13^{12}$
Root discriminant $20.69$
Ramified primes $2, 3, 13$
Class number $3$
Class group $[3]$
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![1, 1, -5, 2, 38, 24, 24, -52, 19, 3, -29, 32, -24, 20, 12, -12, 5, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 5*x^16 - 12*x^15 + 12*x^14 + 20*x^13 - 24*x^12 + 32*x^11 - 29*x^10 + 3*x^9 + 19*x^8 - 52*x^7 + 24*x^6 + 24*x^5 + 38*x^4 + 2*x^3 - 5*x^2 + x + 1)
 
gp: K = bnfinit(x^18 - 3*x^17 + 5*x^16 - 12*x^15 + 12*x^14 + 20*x^13 - 24*x^12 + 32*x^11 - 29*x^10 + 3*x^9 + 19*x^8 - 52*x^7 + 24*x^6 + 24*x^5 + 38*x^4 + 2*x^3 - 5*x^2 + x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 5 x^{16} - 12 x^{15} + 12 x^{14} + 20 x^{13} - 24 x^{12} + 32 x^{11} - 29 x^{10} + 3 x^{9} + 19 x^{8} - 52 x^{7} + 24 x^{6} + 24 x^{5} + 38 x^{4} + 2 x^{3} - 5 x^{2} + x + 1 \)

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:  \(-480852007416803909173248=-\,2^{20}\cdot 3^{9}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.69$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6} a^{12} + \frac{1}{3} a^{11} - \frac{1}{6} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{11} - \frac{1}{3} a^{8} + \frac{1}{6} a^{7} - \frac{1}{3} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{6} a^{14} - \frac{1}{3} a^{11} + \frac{1}{6} a^{10} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} + \frac{1}{6} a^{6} - \frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6}$, $\frac{1}{414} a^{15} - \frac{1}{207} a^{14} + \frac{1}{23} a^{13} + \frac{5}{207} a^{12} + \frac{17}{414} a^{11} + \frac{89}{207} a^{10} - \frac{79}{414} a^{9} + \frac{16}{69} a^{8} + \frac{173}{414} a^{7} + \frac{86}{207} a^{6} + \frac{47}{414} a^{5} - \frac{5}{69} a^{4} + \frac{65}{207} a^{3} + \frac{10}{69} a^{2} - \frac{7}{414} a + \frac{70}{207}$, $\frac{1}{4554} a^{16} - \frac{1}{4554} a^{15} - \frac{61}{2277} a^{14} + \frac{83}{2277} a^{13} - \frac{10}{253} a^{12} + \frac{65}{1518} a^{11} - \frac{248}{759} a^{10} + \frac{1259}{4554} a^{9} - \frac{1525}{4554} a^{8} - \frac{25}{66} a^{7} - \frac{76}{253} a^{6} + \frac{845}{4554} a^{5} - \frac{433}{2277} a^{4} + \frac{1130}{2277} a^{3} + \frac{337}{2277} a^{2} - \frac{419}{4554} a + \frac{1037}{4554}$, $\frac{1}{14805054} a^{17} - \frac{200}{7402527} a^{16} + \frac{3004}{7402527} a^{15} - \frac{1159561}{14805054} a^{14} + \frac{102011}{2467509} a^{13} + \frac{174865}{14805054} a^{12} + \frac{443974}{7402527} a^{11} - \frac{224801}{2467509} a^{10} + \frac{122105}{1345914} a^{9} - \frac{338191}{4935018} a^{8} - \frac{1963802}{7402527} a^{7} + \frac{130463}{822503} a^{6} + \frac{686077}{2467509} a^{5} - \frac{474599}{14805054} a^{4} + \frac{798268}{2467509} a^{3} + \frac{2392501}{14805054} a^{2} + \frac{1032829}{14805054} a - \frac{333596}{7402527}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( \frac{23114}{224319} a^{17} - \frac{314615}{672957} a^{16} + \frac{686800}{672957} a^{15} - \frac{478340}{224319} a^{14} + \frac{2234236}{672957} a^{13} - \frac{149488}{672957} a^{12} - \frac{3527300}{672957} a^{11} + \frac{228880}{29259} a^{10} - \frac{715190}{74773} a^{9} + \frac{4463678}{672957} a^{8} + \frac{148060}{672957} a^{7} - \frac{5662156}{672957} a^{6} + \frac{2771248}{224319} a^{5} - \frac{2586224}{672957} a^{4} + \frac{488248}{224319} a^{3} - \frac{3700996}{672957} a^{2} - \frac{334246}{672957} a + \frac{208286}{224319} \) (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:  \( 128496.88254347249 \)
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

\(\Q(\sqrt{-3}) \), 3.1.676.1, 3.1.2028.1 x3, 6.0.12338352.1, 6.0.12338352.2, 9.1.133451615232.2 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$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$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$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}$
$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}$