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Properties

Label 18.18.1583798252...2569.1
Degree $18$
Signature $[18, 0]$
Discriminant $11^{8}\cdot 43^{14}$
Root discriminant $54.11$
Ramified primes $11, 43$
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![121, 1452, 3454, -11891, -28177, 70453, 24283, -148735, 95171, 35569, -59680, 13898, 8010, -4187, 23, 310, -43, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 - 43*x^16 + 310*x^15 + 23*x^14 - 4187*x^13 + 8010*x^12 + 13898*x^11 - 59680*x^10 + 35569*x^9 + 95171*x^8 - 148735*x^7 + 24283*x^6 + 70453*x^5 - 28177*x^4 - 11891*x^3 + 3454*x^2 + 1452*x + 121)
 
gp: K = bnfinit(x^18 - 5*x^17 - 43*x^16 + 310*x^15 + 23*x^14 - 4187*x^13 + 8010*x^12 + 13898*x^11 - 59680*x^10 + 35569*x^9 + 95171*x^8 - 148735*x^7 + 24283*x^6 + 70453*x^5 - 28177*x^4 - 11891*x^3 + 3454*x^2 + 1452*x + 121, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} - 43 x^{16} + 310 x^{15} + 23 x^{14} - 4187 x^{13} + 8010 x^{12} + 13898 x^{11} - 59680 x^{10} + 35569 x^{9} + 95171 x^{8} - 148735 x^{7} + 24283 x^{6} + 70453 x^{5} - 28177 x^{4} - 11891 x^{3} + 3454 x^{2} + 1452 x + 121 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(15837982522574661576175934102569=11^{8}\cdot 43^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.11$
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:  $11, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{11} a^{14} - \frac{2}{11} a^{13} + \frac{1}{11} a^{12} + \frac{4}{11} a^{11} - \frac{3}{11} a^{10} - \frac{3}{11} a^{9} - \frac{3}{11} a^{8} - \frac{1}{11} a^{6} + \frac{3}{11} a^{5} + \frac{2}{11} a^{4} - \frac{2}{11} a^{3} + \frac{1}{11} a^{2}$, $\frac{1}{22} a^{15} - \frac{3}{22} a^{13} + \frac{3}{11} a^{12} + \frac{5}{22} a^{11} - \frac{9}{22} a^{10} - \frac{9}{22} a^{9} - \frac{3}{11} a^{8} - \frac{1}{22} a^{7} + \frac{1}{22} a^{6} + \frac{4}{11} a^{5} + \frac{1}{11} a^{4} + \frac{4}{11} a^{3} - \frac{9}{22} a^{2} - \frac{1}{2}$, $\frac{1}{2706} a^{16} + \frac{1}{451} a^{15} - \frac{5}{2706} a^{14} - \frac{170}{451} a^{13} + \frac{259}{2706} a^{12} - \frac{69}{902} a^{11} + \frac{449}{2706} a^{10} - \frac{97}{451} a^{9} + \frac{63}{902} a^{8} + \frac{193}{2706} a^{7} - \frac{124}{1353} a^{6} + \frac{18}{41} a^{5} - \frac{100}{451} a^{4} - \frac{1145}{2706} a^{3} - \frac{424}{1353} a^{2} + \frac{91}{246} a - \frac{2}{123}$, $\frac{1}{2594928196454040414} a^{17} + \frac{135558399160043}{2594928196454040414} a^{16} - \frac{7848387190401679}{1297464098227020207} a^{15} - \frac{83960581507917205}{2594928196454040414} a^{14} - \frac{504956693419618384}{1297464098227020207} a^{13} - \frac{421346874359500421}{1297464098227020207} a^{12} - \frac{173974385070972589}{2594928196454040414} a^{11} + \frac{1123756032149354}{10216252741945041} a^{10} - \frac{22824093453839735}{432488032742340069} a^{9} + \frac{222708702614250077}{1297464098227020207} a^{8} - \frac{142012254242102678}{432488032742340069} a^{7} - \frac{4517443095380063}{63290931620830254} a^{6} - \frac{61397076936330919}{432488032742340069} a^{5} - \frac{1003897041036976643}{2594928196454040414} a^{4} + \frac{5720553349716157}{26211395923778186} a^{3} + \frac{36204869249878990}{117951281657001837} a^{2} + \frac{10418758478526071}{21445687574000334} a - \frac{4722034540754023}{21445687574000334}$

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:  $17$
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:  \( 9042270276.65 \) (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

3.3.1849.1, 3.3.473.1, 6.6.413674921.1, 9.9.361790571383417.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$43$43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.3.2.1$x^{3} - 43$$3$$1$$2$$C_3$$[\ ]_{3}$
43.6.5.1$x^{6} - 43$$6$$1$$5$$C_6$$[\ ]_{6}$
43.6.5.1$x^{6} - 43$$6$$1$$5$$C_6$$[\ ]_{6}$