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Properties

Label 18.18.2778760824...0000.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{16}\cdot 5^{8}\cdot 17^{4}\cdot 37^{9}$
Root discriminant $43.23$
Ramified primes $2, 5, 17, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\wr C_3:C_2$ (as 18T88)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6047, -5051, -36045, 34932, 77066, -86246, -73600, 101212, 28339, -60741, 757, 18208, -3140, -2454, 618, 144, -43, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 43*x^16 + 144*x^15 + 618*x^14 - 2454*x^13 - 3140*x^12 + 18208*x^11 + 757*x^10 - 60741*x^9 + 28339*x^8 + 101212*x^7 - 73600*x^6 - 86246*x^5 + 77066*x^4 + 34932*x^3 - 36045*x^2 - 5051*x + 6047)
 
gp: K = bnfinit(x^18 - 3*x^17 - 43*x^16 + 144*x^15 + 618*x^14 - 2454*x^13 - 3140*x^12 + 18208*x^11 + 757*x^10 - 60741*x^9 + 28339*x^8 + 101212*x^7 - 73600*x^6 - 86246*x^5 + 77066*x^4 + 34932*x^3 - 36045*x^2 - 5051*x + 6047, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 43 x^{16} + 144 x^{15} + 618 x^{14} - 2454 x^{13} - 3140 x^{12} + 18208 x^{11} + 757 x^{10} - 60741 x^{9} + 28339 x^{8} + 101212 x^{7} - 73600 x^{6} - 86246 x^{5} + 77066 x^{4} + 34932 x^{3} - 36045 x^{2} - 5051 x + 6047 \)

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:  \(277876082417270428595200000000=2^{16}\cdot 5^{8}\cdot 17^{4}\cdot 37^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.23$
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:  $2, 5, 17, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{15} + \frac{1}{10} a^{14} - \frac{1}{10} a^{13} + \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{10} a^{3} - \frac{1}{10} a^{2} - \frac{3}{10} a + \frac{2}{5}$, $\frac{1}{310} a^{16} - \frac{4}{155} a^{15} - \frac{3}{62} a^{14} - \frac{13}{155} a^{13} - \frac{33}{310} a^{12} + \frac{69}{310} a^{11} + \frac{11}{310} a^{10} - \frac{97}{310} a^{9} + \frac{59}{155} a^{8} - \frac{151}{310} a^{7} + \frac{119}{310} a^{6} + \frac{1}{62} a^{5} - \frac{33}{310} a^{4} - \frac{17}{62} a^{3} + \frac{23}{155} a^{2} + \frac{41}{310} a + \frac{119}{310}$, $\frac{1}{3924321295140559185430} a^{17} - \frac{186401581383088565}{392432129514055918543} a^{16} - \frac{34288753785254140765}{784864259028111837086} a^{15} - \frac{3056606626672763536}{63295504760331599765} a^{14} + \frac{21465940378731955546}{392432129514055918543} a^{13} - \frac{168334960006800398583}{784864259028111837086} a^{12} + \frac{751474384247564308251}{3924321295140559185430} a^{11} + \frac{565037836731341847359}{3924321295140559185430} a^{10} + \frac{910799031528560130674}{1962160647570279592715} a^{9} - \frac{908346177298275841498}{1962160647570279592715} a^{8} - \frac{1364193130820775523783}{3924321295140559185430} a^{7} - \frac{632906157167017955493}{3924321295140559185430} a^{6} + \frac{1150473353331922177023}{3924321295140559185430} a^{5} - \frac{6283353672264645129}{392432129514055918543} a^{4} + \frac{179912420150092296767}{392432129514055918543} a^{3} - \frac{185528950311876271855}{784864259028111837086} a^{2} + \frac{71605218453561885955}{392432129514055918543} a + \frac{1613207065885998195893}{3924321295140559185430}$

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:  \( 957797101.545 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\wr C_3:C_2$ (as 18T88):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 162
The 13 conjugacy class representatives for $C_3\wr C_3:C_2$
Character table for $C_3\wr C_3:C_2$

Intermediate fields

\(\Q(\sqrt{37}) \), 3.3.148.1 x3, 6.6.810448.1, 9.9.86661204640000.1 x3

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

Sibling fields

Degree 9 siblings: data not computed
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 ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
2Data not computed
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$17$17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
37Data not computed