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Properties

Label 18.6.18877164575...4001.1
Degree $18$
Signature $[6, 6]$
Discriminant $19^{12}\cdot 31^{8}$
Root discriminant $32.76$
Ramified primes $19, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3\times A_4$ (as 18T31)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![88, -276, -76, 973, -982, -1182, 1614, -25, -1136, -375, -715, -499, -248, -160, -37, -4, 7, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 7*x^16 - 4*x^15 - 37*x^14 - 160*x^13 - 248*x^12 - 499*x^11 - 715*x^10 - 375*x^9 - 1136*x^8 - 25*x^7 + 1614*x^6 - 1182*x^5 - 982*x^4 + 973*x^3 - 76*x^2 - 276*x + 88)
 
gp: K = bnfinit(x^18 + 7*x^16 - 4*x^15 - 37*x^14 - 160*x^13 - 248*x^12 - 499*x^11 - 715*x^10 - 375*x^9 - 1136*x^8 - 25*x^7 + 1614*x^6 - 1182*x^5 - 982*x^4 + 973*x^3 - 76*x^2 - 276*x + 88, 1)
 

Normalized defining polynomial

\( x^{18} + 7 x^{16} - 4 x^{15} - 37 x^{14} - 160 x^{13} - 248 x^{12} - 499 x^{11} - 715 x^{10} - 375 x^{9} - 1136 x^{8} - 25 x^{7} + 1614 x^{6} - 1182 x^{5} - 982 x^{4} + 973 x^{3} - 76 x^{2} - 276 x + 88 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1887716457505981006207134001=19^{12}\cdot 31^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.76$
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:  $19, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{12} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{6} a^{8} + \frac{1}{3} a^{7} + \frac{1}{6} a^{6} + \frac{5}{12} a^{5} - \frac{5}{12} a^{3} - \frac{1}{12} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{12} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{6} a^{7} + \frac{5}{12} a^{6} - \frac{5}{12} a^{4} - \frac{1}{12} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6} a$, $\frac{1}{12} a^{14} - \frac{1}{4} a^{11} + \frac{1}{12} a^{10} + \frac{1}{12} a^{9} + \frac{1}{6} a^{8} + \frac{5}{12} a^{7} - \frac{1}{2} a^{6} - \frac{1}{6} a^{5} - \frac{1}{12} a^{4} + \frac{5}{12} a^{3} - \frac{5}{12} a^{2} - \frac{1}{2} a$, $\frac{1}{36} a^{15} - \frac{1}{36} a^{13} + \frac{1}{36} a^{12} - \frac{1}{18} a^{11} - \frac{1}{18} a^{10} - \frac{1}{18} a^{9} - \frac{7}{36} a^{8} - \frac{4}{9} a^{7} + \frac{13}{36} a^{6} - \frac{17}{36} a^{5} + \frac{4}{9} a^{4} + \frac{1}{6} a^{3} + \frac{1}{9} a + \frac{1}{9}$, $\frac{1}{36} a^{16} - \frac{1}{36} a^{14} + \frac{1}{36} a^{13} + \frac{1}{36} a^{12} - \frac{1}{18} a^{11} + \frac{7}{36} a^{10} + \frac{1}{18} a^{9} - \frac{1}{9} a^{8} - \frac{11}{36} a^{7} - \frac{11}{36} a^{6} - \frac{5}{36} a^{5} + \frac{1}{6} a^{4} + \frac{1}{12} a^{3} - \frac{17}{36} a^{2} + \frac{5}{18} a + \frac{1}{3}$, $\frac{1}{17954555432716115273808} a^{17} + \frac{214815823210120822511}{17954555432716115273808} a^{16} + \frac{5589328709640069803}{2244319429089514409226} a^{15} - \frac{154571563858515812395}{4488638858179028818452} a^{14} + \frac{359124526145130134695}{17954555432716115273808} a^{13} - \frac{9780154259630014025}{1632232312065101388528} a^{12} + \frac{672362438140161695591}{17954555432716115273808} a^{11} + \frac{473281841267515664599}{2992425905452685878968} a^{10} - \frac{686091085491221995957}{17954555432716115273808} a^{9} + \frac{174321471196614930605}{997475301817561959656} a^{8} + \frac{4225236961914794961023}{8977277716358057636904} a^{7} + \frac{3386016058296729164129}{17954555432716115273808} a^{6} + \frac{6686697142336648697065}{17954555432716115273808} a^{5} + \frac{3931299075503156916677}{17954555432716115273808} a^{4} - \frac{8431491558611318711519}{17954555432716115273808} a^{3} - \frac{221774501794517891293}{498737650908780979828} a^{2} - \frac{1835711963937129245903}{4488638858179028818452} a - \frac{26958166716350856583}{204029039008137673566}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 9826413.58709 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times A_4$ (as 18T31):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 12 conjugacy class representatives for $S_3\times A_4$
Character table for $S_3\times A_4$

Intermediate fields

3.3.361.1, 3.1.31.1, 6.6.125238481.1, 9.3.1401543840871.1

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

Sibling fields

Degree 12 sibling: data not computed
Degree 18 sibling: 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$19$19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
31Data not computed