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Properties

Label 18.0.10591107618...0000.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{25}\cdot 5^{15}$
Root discriminant $27.91$
Ramified primes $2, 3, 5$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\times C_3^2:S_3$ (as 18T52)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1024, -2048, 9344, -15392, 21040, -20168, 16596, -10288, 5604, -2415, 1049, -388, 204, -122, 90, -52, 24, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 24*x^16 - 52*x^15 + 90*x^14 - 122*x^13 + 204*x^12 - 388*x^11 + 1049*x^10 - 2415*x^9 + 5604*x^8 - 10288*x^7 + 16596*x^6 - 20168*x^5 + 21040*x^4 - 15392*x^3 + 9344*x^2 - 2048*x + 1024)
 
gp: K = bnfinit(x^18 - 7*x^17 + 24*x^16 - 52*x^15 + 90*x^14 - 122*x^13 + 204*x^12 - 388*x^11 + 1049*x^10 - 2415*x^9 + 5604*x^8 - 10288*x^7 + 16596*x^6 - 20168*x^5 + 21040*x^4 - 15392*x^3 + 9344*x^2 - 2048*x + 1024, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 24 x^{16} - 52 x^{15} + 90 x^{14} - 122 x^{13} + 204 x^{12} - 388 x^{11} + 1049 x^{10} - 2415 x^{9} + 5604 x^{8} - 10288 x^{7} + 16596 x^{6} - 20168 x^{5} + 21040 x^{4} - 15392 x^{3} + 9344 x^{2} - 2048 x + 1024 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-105911076180375000000000000=-\,2^{12}\cdot 3^{25}\cdot 5^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.91$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{9} + \frac{1}{8} a^{5} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} - \frac{1}{16} a^{7} - \frac{3}{16} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{10} - \frac{1}{16} a^{8} + \frac{1}{16} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{12} - \frac{1}{32} a^{11} - \frac{1}{32} a^{10} + \frac{1}{32} a^{9} + \frac{1}{32} a^{8} + \frac{1}{32} a^{7} - \frac{3}{32} a^{6} + \frac{1}{16} a^{5} + \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{14} - \frac{1}{64} a^{13} + \frac{1}{64} a^{12} - \frac{1}{64} a^{11} - \frac{1}{64} a^{10} - \frac{3}{64} a^{9} - \frac{1}{64} a^{8} - \frac{3}{64} a^{7} + \frac{1}{16} a^{6} - \frac{1}{4} a^{5} - \frac{1}{16} a^{4} - \frac{3}{8} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{64} a^{15} - \frac{1}{32} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} + \frac{1}{64} a^{7} + \frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{3}{16} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4352} a^{16} - \frac{15}{2176} a^{15} + \frac{13}{2176} a^{14} - \frac{1}{2176} a^{13} - \frac{9}{544} a^{12} - \frac{41}{2176} a^{11} + \frac{109}{2176} a^{10} - \frac{61}{2176} a^{9} - \frac{65}{4352} a^{8} + \frac{7}{544} a^{7} - \frac{49}{1088} a^{6} + \frac{177}{1088} a^{5} - \frac{37}{544} a^{4} - \frac{135}{272} a^{3} - \frac{45}{136} a^{2} - \frac{7}{34} a - \frac{8}{17}$, $\frac{1}{4740297525564928} a^{17} + \frac{32721454571}{592537190695616} a^{16} - \frac{3975586057869}{2370148762782464} a^{15} + \frac{6162180923661}{2370148762782464} a^{14} + \frac{4964891791451}{1185074381391232} a^{13} - \frac{34487323484977}{2370148762782464} a^{12} - \frac{32540575108281}{2370148762782464} a^{11} + \frac{116725566584913}{2370148762782464} a^{10} - \frac{289087709437213}{4740297525564928} a^{9} - \frac{144609389671375}{2370148762782464} a^{8} + \frac{25413642637947}{1185074381391232} a^{7} + \frac{63837298316251}{1185074381391232} a^{6} - \frac{66262917776581}{296268595347808} a^{5} + \frac{21413047799469}{148134297673904} a^{4} + \frac{32051030698623}{74067148836952} a^{3} + \frac{34127292862955}{74067148836952} a^{2} - \frac{1253600929117}{18516787209238} a - \frac{4214905390593}{9258393604619}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}$, which has order $2$ (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:  $8$
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:  \( 9396088.902996559 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3^2:S_3$ (as 18T52):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times C_3^2:S_3$
Character table for $C_2\times C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-15}) \), 3.1.300.1, 6.0.1350000.1, 9.3.177147000000.1

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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.12.22.83$x^{12} + 27 x^{11} - 27 x^{10} + 3 x^{9} - 9 x^{8} - 36 x^{7} - 9 x^{6} + 27 x^{5} + 36 x^{3} + 27 x + 36$$6$$2$$22$$C_6\times S_3$$[5/2]_{2}^{6}$
5Data not computed