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Properties

Label 18.6.77807296868...3696.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{30}\cdot 3^{24}\cdot 37^{6}$
Root discriminant $45.77$
Ramified primes $2, 3, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3:S_3:S_4$ (as 18T154)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-128, -576, 1440, 13632, 31944, 27936, -6671, -29208, -12690, 7744, 6225, -1152, -1604, 108, 261, -4, -24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 24*x^16 - 4*x^15 + 261*x^14 + 108*x^13 - 1604*x^12 - 1152*x^11 + 6225*x^10 + 7744*x^9 - 12690*x^8 - 29208*x^7 - 6671*x^6 + 27936*x^5 + 31944*x^4 + 13632*x^3 + 1440*x^2 - 576*x - 128)
 
gp: K = bnfinit(x^18 - 24*x^16 - 4*x^15 + 261*x^14 + 108*x^13 - 1604*x^12 - 1152*x^11 + 6225*x^10 + 7744*x^9 - 12690*x^8 - 29208*x^7 - 6671*x^6 + 27936*x^5 + 31944*x^4 + 13632*x^3 + 1440*x^2 - 576*x - 128, 1)
 

Normalized defining polynomial

\( x^{18} - 24 x^{16} - 4 x^{15} + 261 x^{14} + 108 x^{13} - 1604 x^{12} - 1152 x^{11} + 6225 x^{10} + 7744 x^{9} - 12690 x^{8} - 29208 x^{7} - 6671 x^{6} + 27936 x^{5} + 31944 x^{4} + 13632 x^{3} + 1440 x^{2} - 576 x - 128 \)

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:  \(778072968681250317161459613696=2^{30}\cdot 3^{24}\cdot 37^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $45.77$
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:  $2, 3, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{138} a^{13} + \frac{2}{23} a^{12} - \frac{1}{69} a^{11} + \frac{1}{23} a^{10} - \frac{7}{138} a^{9} + \frac{14}{69} a^{8} - \frac{5}{69} a^{7} - \frac{11}{69} a^{6} - \frac{1}{2} a^{5} + \frac{13}{69} a^{4} - \frac{3}{23} a^{3} + \frac{28}{69} a^{2} - \frac{5}{138} a + \frac{20}{69}$, $\frac{1}{6348} a^{14} - \frac{3}{1058} a^{13} + \frac{81}{529} a^{12} - \frac{6}{529} a^{11} + \frac{733}{6348} a^{10} - \frac{295}{3174} a^{9} - \frac{385}{1587} a^{8} + \frac{196}{1587} a^{7} - \frac{2675}{6348} a^{6} + \frac{841}{3174} a^{5} - \frac{179}{1058} a^{4} + \frac{42}{529} a^{3} + \frac{1765}{6348} a^{2} + \frac{877}{3174} a - \frac{139}{1587}$, $\frac{1}{1460040} a^{15} + \frac{3}{48668} a^{14} - \frac{1301}{365010} a^{13} - \frac{4483}{121670} a^{12} - \frac{125539}{1460040} a^{11} - \frac{74977}{730020} a^{10} + \frac{24376}{182505} a^{9} + \frac{32737}{182505} a^{8} - \frac{218829}{486680} a^{7} - \frac{280091}{730020} a^{6} - \frac{25977}{243340} a^{5} + \frac{161039}{365010} a^{4} + \frac{37153}{1460040} a^{3} - \frac{16523}{146004} a^{2} - \frac{2643}{5290} a - \frac{21546}{60835}$, $\frac{1}{67161840} a^{16} + \frac{7}{33580920} a^{15} - \frac{61}{2798410} a^{14} + \frac{37817}{16790460} a^{13} - \frac{353683}{67161840} a^{12} + \frac{1057597}{6716184} a^{11} - \frac{1855187}{16790460} a^{10} - \frac{205027}{4197615} a^{9} - \frac{15926543}{67161840} a^{8} - \frac{1148101}{6716184} a^{7} + \frac{32537}{2238728} a^{6} + \frac{1915419}{5596820} a^{5} + \frac{20020577}{67161840} a^{4} - \frac{1799483}{11193640} a^{3} + \frac{1561819}{8395230} a^{2} + \frac{831041}{2798410} a - \frac{1926521}{4197615}$, $\frac{1}{3089444640} a^{17} - \frac{1}{193090290} a^{16} + \frac{29}{386180580} a^{15} - \frac{8421}{257453720} a^{14} - \frac{2381049}{1029814880} a^{13} - \frac{34082323}{257453720} a^{12} + \frac{37677349}{257453720} a^{11} - \frac{6819269}{128726860} a^{10} - \frac{58464979}{617888928} a^{9} - \frac{19093733}{64363430} a^{8} - \frac{3839029}{22387280} a^{7} + \frac{18082741}{193090290} a^{6} + \frac{93114509}{617888928} a^{5} + \frac{48217549}{96545145} a^{4} + \frac{3683581}{77236116} a^{3} - \frac{103091027}{386180580} a^{2} + \frac{27728971}{193090290} a + \frac{1761173}{96545145}$

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:  \( 439659323.38 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3:S_4$ (as 18T154):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 20 conjugacy class representatives for $C_3:S_3:S_4$
Character table for $C_3:S_3:S_4$

Intermediate fields

3.3.148.1, 6.2.87616.1, 9.9.220521111330816.1

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

Sibling fields

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 ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.6.0.1}{6} }$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.6.6.7$x^{6} + 2 x^{2} + 2 x + 2$$6$$1$$6$$S_4$$[4/3, 4/3]_{3}^{2}$
2.12.24.342$x^{12} + 4 x^{11} + 2 x^{10} + 4 x^{9} + 4 x^{5} - 2 x^{4} + 4 x^{3} - 2 x^{2} + 4 x - 2$$12$$1$$24$$C_2 \times S_4$$[4/3, 4/3, 3]_{3}^{2}$
$3$3.9.12.17$x^{9} + 6 x^{8} + 6 x^{6} + 27$$3$$3$$12$$C_3^2 : S_3 $$[2, 2]^{6}$
3.9.12.17$x^{9} + 6 x^{8} + 6 x^{6} + 27$$3$$3$$12$$C_3^2 : S_3 $$[2, 2]^{6}$
$37$$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{37}$$x + 2$$1$$1$$0$Trivial$[\ ]$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
37.4.2.1$x^{4} + 333 x^{2} + 34225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$