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Properties

Label 18.0.20352109818...8608.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 41^{6}$
Root discriminant $19.72$
Ramified primes $2, 3, 41$
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![343, 0, 0, -136, 0, 0, 435, 0, 0, 4, 0, 0, -15, 0, 0, -4, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^15 - 15*x^12 + 4*x^9 + 435*x^6 - 136*x^3 + 343)
 
gp: K = bnfinit(x^18 - 4*x^15 - 15*x^12 + 4*x^9 + 435*x^6 - 136*x^3 + 343, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{15} - 15 x^{12} + 4 x^{9} + 435 x^{6} - 136 x^{3} + 343 \)

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:  \(-203521098186458046148608=-\,2^{12}\cdot 3^{21}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.72$
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:  $2, 3, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{6} a^{9} + \frac{1}{6} a^{6} + \frac{1}{6} a^{3} + \frac{1}{6}$, $\frac{1}{6} a^{10} + \frac{1}{6} a^{7} + \frac{1}{6} a^{4} + \frac{1}{6} a$, $\frac{1}{6} a^{11} + \frac{1}{6} a^{8} + \frac{1}{6} a^{5} + \frac{1}{6} a^{2}$, $\frac{1}{60} a^{12} - \frac{1}{30} a^{9} + \frac{1}{15} a^{6} + \frac{4}{15} a^{3} + \frac{3}{20}$, $\frac{1}{180} a^{13} - \frac{1}{18} a^{11} - \frac{1}{90} a^{10} + \frac{1}{18} a^{9} - \frac{7}{18} a^{8} + \frac{1}{45} a^{7} + \frac{7}{18} a^{6} + \frac{5}{18} a^{5} - \frac{11}{45} a^{4} - \frac{5}{18} a^{3} - \frac{1}{18} a^{2} + \frac{1}{20} a + \frac{1}{18}$, $\frac{1}{180} a^{14} - \frac{1}{180} a^{12} - \frac{1}{90} a^{11} + \frac{1}{18} a^{10} + \frac{1}{90} a^{9} + \frac{1}{45} a^{8} + \frac{7}{18} a^{7} - \frac{1}{45} a^{6} - \frac{11}{45} a^{5} - \frac{5}{18} a^{4} + \frac{11}{45} a^{3} + \frac{1}{20} a^{2} + \frac{1}{18} a - \frac{1}{20}$, $\frac{1}{114300} a^{15} + \frac{139}{114300} a^{12} - \frac{1499}{57150} a^{9} - \frac{1052}{5715} a^{6} + \frac{217}{4572} a^{3} - \frac{28271}{114300}$, $\frac{1}{800100} a^{16} - \frac{377}{266700} a^{13} - \frac{1}{18} a^{11} + \frac{14173}{200025} a^{10} + \frac{1}{18} a^{9} - \frac{7}{18} a^{8} + \frac{37393}{80010} a^{7} + \frac{7}{18} a^{6} + \frac{5}{18} a^{5} + \frac{7897}{53340} a^{4} - \frac{5}{18} a^{3} - \frac{1}{18} a^{2} - \frac{211151}{800100} a + \frac{1}{18}$, $\frac{1}{16802100} a^{17} - \frac{1}{2400300} a^{16} + \frac{1}{342900} a^{15} - \frac{16123}{8401050} a^{14} + \frac{1394}{600075} a^{13} + \frac{139}{342900} a^{12} - \frac{607289}{8401050} a^{11} + \frac{16942}{600075} a^{10} - \frac{1499}{171450} a^{9} + \frac{51617}{1680210} a^{8} + \frac{5773}{24003} a^{7} - \frac{1052}{17145} a^{6} - \frac{609277}{3360420} a^{5} + \frac{123883}{480060} a^{4} + \frac{4789}{13716} a^{3} + \frac{3088157}{8401050} a^{2} - \frac{207797}{1200150} a - \frac{142571}{342900}$

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:  \( \frac{1}{450} a^{15} - \frac{7}{900} a^{12} - \frac{13}{450} a^{9} - \frac{1}{45} a^{6} + \frac{89}{90} a^{3} + \frac{293}{900} \) (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:  \( 132908.71058495017 \)
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

\(\Q(\sqrt{-3}) \), 3.1.108.1 x3, 3.1.492.1, 6.0.726192.1, 6.0.34992.1, 9.1.260461832256.1 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.12.14.6$x^{12} + 3 x^{11} + 3 x^{10} - 6 x^{9} + 3 x^{8} + 9 x^{7} + 9 x^{4} + 9 x^{3} + 9$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
$41$41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$