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Properties

Label 18.8.56986226150...6707.1
Degree $18$
Signature $[8, 5]$
Discriminant $-\,3^{27}\cdot 73^{3}\cdot 577^{3}$
Root discriminant $30.65$
Ramified primes $3, 73, 577$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T765

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-107, 1230, -2631, 948, 4857, -7284, 384, 6909, -4908, -1109, 2739, -1041, -138, 300, -168, 48, 3, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 3*x^16 + 48*x^15 - 168*x^14 + 300*x^13 - 138*x^12 - 1041*x^11 + 2739*x^10 - 1109*x^9 - 4908*x^8 + 6909*x^7 + 384*x^6 - 7284*x^5 + 4857*x^4 + 948*x^3 - 2631*x^2 + 1230*x - 107)
 
gp: K = bnfinit(x^18 - 6*x^17 + 3*x^16 + 48*x^15 - 168*x^14 + 300*x^13 - 138*x^12 - 1041*x^11 + 2739*x^10 - 1109*x^9 - 4908*x^8 + 6909*x^7 + 384*x^6 - 7284*x^5 + 4857*x^4 + 948*x^3 - 2631*x^2 + 1230*x - 107, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 3 x^{16} + 48 x^{15} - 168 x^{14} + 300 x^{13} - 138 x^{12} - 1041 x^{11} + 2739 x^{10} - 1109 x^{9} - 4908 x^{8} + 6909 x^{7} + 384 x^{6} - 7284 x^{5} + 4857 x^{4} + 948 x^{3} - 2631 x^{2} + 1230 x - 107 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-569862261508654647313096707=-\,3^{27}\cdot 73^{3}\cdot 577^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.65$
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:  $3, 73, 577$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{9} - \frac{1}{9} a^{3} + \frac{1}{9}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{10} - \frac{1}{9} a^{4} + \frac{1}{9} a$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{11} - \frac{1}{9} a^{5} + \frac{1}{9} a^{2}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{9} - \frac{1}{9} a^{6} + \frac{1}{9}$, $\frac{1}{9} a^{16} - \frac{1}{9} a^{10} - \frac{1}{9} a^{7} + \frac{1}{9} a$, $\frac{1}{8163877252191471160263} a^{17} + \frac{33563428643553699802}{8163877252191471160263} a^{16} + \frac{134017293281731316851}{2721292417397157053421} a^{15} - \frac{41285159947703843143}{907097472465719017807} a^{14} - \frac{413914239849579026965}{8163877252191471160263} a^{13} - \frac{190350607724188237339}{8163877252191471160263} a^{12} - \frac{737067270899417210791}{8163877252191471160263} a^{11} + \frac{376298054499842762575}{2721292417397157053421} a^{10} - \frac{275031140653227092543}{8163877252191471160263} a^{9} + \frac{62387760332871957272}{8163877252191471160263} a^{8} - \frac{1183847437983052846528}{8163877252191471160263} a^{7} + \frac{450095302007518741727}{2721292417397157053421} a^{6} + \frac{178503643355602853399}{2721292417397157053421} a^{5} - \frac{2552481043858523756696}{8163877252191471160263} a^{4} - \frac{23052493803176244335}{8163877252191471160263} a^{3} - \frac{2356858210156793848742}{8163877252191471160263} a^{2} - \frac{750751735640854636208}{2721292417397157053421} a - \frac{2542610716671783870946}{8163877252191471160263}$

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:  $12$
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:  \( 3198073.57437 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T765:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 110 conjugacy class representatives for t18n765 are not computed
Character table for t18n765 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.9.22384826361.1

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 $18$ R $18$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
3Data not computed
$73$$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.2.0.1$x^{2} - x + 11$$1$$2$$0$$C_2$$[\ ]^{2}$
73.3.0.1$x^{3} - x + 14$$1$$3$$0$$C_3$$[\ ]^{3}$
73.3.0.1$x^{3} - x + 14$$1$$3$$0$$C_3$$[\ ]^{3}$
73.4.3.2$x^{4} - 1825$$4$$1$$3$$C_4$$[\ ]_{4}$
577Data not computed