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Properties

Label 18.14.6489745507...0553.1
Degree $18$
Signature $[14, 2]$
Discriminant $3^{24}\cdot 73^{5}\cdot 577^{4}$
Root discriminant $58.53$
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![116299, 939309, 605688, -1283994, -323886, 593118, -310223, 21000, 175299, -60954, -12687, 9960, -5787, -306, 1101, -1, -60, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 60*x^16 - x^15 + 1101*x^14 - 306*x^13 - 5787*x^12 + 9960*x^11 - 12687*x^10 - 60954*x^9 + 175299*x^8 + 21000*x^7 - 310223*x^6 + 593118*x^5 - 323886*x^4 - 1283994*x^3 + 605688*x^2 + 939309*x + 116299)
 
gp: K = bnfinit(x^18 - 60*x^16 - x^15 + 1101*x^14 - 306*x^13 - 5787*x^12 + 9960*x^11 - 12687*x^10 - 60954*x^9 + 175299*x^8 + 21000*x^7 - 310223*x^6 + 593118*x^5 - 323886*x^4 - 1283994*x^3 + 605688*x^2 + 939309*x + 116299, 1)
 

Normalized defining polynomial

\( x^{18} - 60 x^{16} - x^{15} + 1101 x^{14} - 306 x^{13} - 5787 x^{12} + 9960 x^{11} - 12687 x^{10} - 60954 x^{9} + 175299 x^{8} + 21000 x^{7} - 310223 x^{6} + 593118 x^{5} - 323886 x^{4} - 1283994 x^{3} + 605688 x^{2} + 939309 x + 116299 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(64897455079312633154135966180553=3^{24}\cdot 73^{5}\cdot 577^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{424341492524621618115230935776010019944325071349559341} a^{17} + \frac{141263361455783321295599586833380895465519038298517304}{424341492524621618115230935776010019944325071349559341} a^{16} + \frac{140884384320133400544047180545101571638995350253233159}{424341492524621618115230935776010019944325071349559341} a^{15} - \frac{12980744165595299048002694984211344617233728663074991}{424341492524621618115230935776010019944325071349559341} a^{14} - \frac{163154792044419852220281539268288138074516654564164035}{424341492524621618115230935776010019944325071349559341} a^{13} - \frac{126553281925417978395043699036292032053767507105651850}{424341492524621618115230935776010019944325071349559341} a^{12} - \frac{180426710292964217312660714092597860248510570810053471}{424341492524621618115230935776010019944325071349559341} a^{11} + \frac{19117706321380336055998997508954516856039349910829544}{424341492524621618115230935776010019944325071349559341} a^{10} - \frac{82689122039574744229738365351871953672581191518398041}{424341492524621618115230935776010019944325071349559341} a^{9} - \frac{81646115306683571579552466709247007128452298451085263}{424341492524621618115230935776010019944325071349559341} a^{8} - \frac{9527636623571166754236386367485552739510908968131335}{424341492524621618115230935776010019944325071349559341} a^{7} - \frac{45983938318908681953361007915730196234483299402453371}{424341492524621618115230935776010019944325071349559341} a^{6} + \frac{168797148881088152766443103445716266629061404632355889}{424341492524621618115230935776010019944325071349559341} a^{5} - \frac{195528956224091929626682474460129888587411866928730252}{424341492524621618115230935776010019944325071349559341} a^{4} + \frac{143775460770364833707350107680170456273654917770008638}{424341492524621618115230935776010019944325071349559341} a^{3} + \frac{125436458036434932921088341834503812940738381103943270}{424341492524621618115230935776010019944325071349559341} a^{2} - \frac{125684322875370414865585072729168546271021032918731489}{424341492524621618115230935776010019944325071349559341} a - \frac{150361358526317787563491638681135831042895752551159268}{424341492524621618115230935776010019944325071349559341}$

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:  $15$
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:  \( 5487619461.51 \) (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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R $18$ $18$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ $18$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }$ $18$ $18$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$73$$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$
73.4.2.1$x^{4} + 1533 x^{2} + 644809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
73.4.2.1$x^{4} + 1533 x^{2} + 644809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
73.6.0.1$x^{6} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
577Data not computed