↑

Properties

Label 18.0.31516489264...9808.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{33}\cdot 7^{12}$
Root discriminant $43.53$
Ramified primes $2, 3, 7$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![484, -396, -2448, 8688, 5652, -25938, 29967, -22095, 12681, -5320, 1413, -315, 318, -369, 279, -138, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 138*x^15 + 279*x^14 - 369*x^13 + 318*x^12 - 315*x^11 + 1413*x^10 - 5320*x^9 + 12681*x^8 - 22095*x^7 + 29967*x^6 - 25938*x^5 + 5652*x^4 + 8688*x^3 - 2448*x^2 - 396*x + 484)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 - 138*x^15 + 279*x^14 - 369*x^13 + 318*x^12 - 315*x^11 + 1413*x^10 - 5320*x^9 + 12681*x^8 - 22095*x^7 + 29967*x^6 - 25938*x^5 + 5652*x^4 + 8688*x^3 - 2448*x^2 - 396*x + 484, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} - 138 x^{15} + 279 x^{14} - 369 x^{13} + 318 x^{12} - 315 x^{11} + 1413 x^{10} - 5320 x^{9} + 12681 x^{8} - 22095 x^{7} + 29967 x^{6} - 25938 x^{5} + 5652 x^{4} + 8688 x^{3} - 2448 x^{2} - 396 x + 484 \)

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:  \(-315164892649262158461997559808=-\,2^{12}\cdot 3^{33}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.53$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{2}{9} a^{5} - \frac{2}{9} a^{4} + \frac{2}{9} a^{3} + \frac{4}{9} a^{2} - \frac{4}{9} a + \frac{4}{9}$, $\frac{1}{45} a^{9} - \frac{1}{45} a^{8} + \frac{1}{45} a^{7} + \frac{2}{45} a^{6} + \frac{7}{45} a^{5} - \frac{16}{45} a^{4} + \frac{13}{45} a^{3} + \frac{14}{45} a^{2} - \frac{1}{9} a + \frac{1}{5}$, $\frac{1}{45} a^{10} + \frac{1}{15} a^{7} - \frac{2}{15} a^{6} - \frac{1}{5} a^{5} - \frac{1}{15} a^{4} - \frac{1}{15} a^{3} + \frac{1}{5} a^{2} + \frac{4}{45} a - \frac{2}{15}$, $\frac{1}{45} a^{11} - \frac{2}{45} a^{8} - \frac{1}{45} a^{7} + \frac{1}{45} a^{6} - \frac{13}{45} a^{5} + \frac{7}{45} a^{4} - \frac{16}{45} a^{3} - \frac{16}{45} a^{2} + \frac{14}{45} a - \frac{1}{9}$, $\frac{1}{270} a^{12} + \frac{1}{270} a^{9} + \frac{1}{45} a^{8} - \frac{1}{45} a^{7} + \frac{11}{90} a^{6} + \frac{8}{45} a^{5} - \frac{14}{45} a^{4} - \frac{77}{270} a^{3} + \frac{1}{45} a^{2} + \frac{4}{9} a - \frac{19}{135}$, $\frac{1}{540} a^{13} + \frac{1}{540} a^{10} + \frac{1}{20} a^{7} + \frac{1}{15} a^{6} + \frac{4}{15} a^{5} + \frac{19}{540} a^{4} - \frac{2}{15} a^{3} + \frac{1}{15} a^{2} + \frac{131}{270} a + \frac{2}{5}$, $\frac{1}{2700} a^{14} + \frac{1}{1350} a^{13} + \frac{1}{1350} a^{12} + \frac{1}{108} a^{11} - \frac{1}{270} a^{10} + \frac{13}{1350} a^{9} + \frac{1}{100} a^{8} - \frac{1}{150} a^{7} - \frac{1}{50} a^{6} - \frac{593}{2700} a^{5} + \frac{289}{1350} a^{4} + \frac{451}{1350} a^{3} - \frac{11}{270} a^{2} - \frac{313}{675} a + \frac{56}{675}$, $\frac{1}{2700} a^{15} - \frac{1}{1350} a^{13} + \frac{1}{2700} a^{12} - \frac{7}{1350} a^{10} + \frac{1}{180} a^{9} - \frac{2}{75} a^{8} - \frac{7}{50} a^{7} - \frac{61}{540} a^{6} - \frac{31}{75} a^{5} + \frac{233}{1350} a^{4} + \frac{113}{1350} a^{3} - \frac{17}{75} a^{2} - \frac{143}{675} a - \frac{49}{225}$, $\frac{1}{16200} a^{16} + \frac{1}{16200} a^{15} - \frac{7}{16200} a^{13} - \frac{1}{648} a^{12} + \frac{1}{450} a^{11} - \frac{149}{16200} a^{10} + \frac{17}{3240} a^{9} + \frac{7}{225} a^{8} - \frac{449}{16200} a^{7} - \frac{2519}{16200} a^{6} + \frac{139}{450} a^{5} - \frac{791}{8100} a^{4} + \frac{646}{2025} a^{3} + \frac{16}{75} a^{2} - \frac{733}{2025} a - \frac{223}{810}$, $\frac{1}{412843959000} a^{17} - \frac{1391279}{206421979500} a^{16} - \frac{55112903}{412843959000} a^{15} - \frac{52259851}{412843959000} a^{14} + \frac{166493879}{206421979500} a^{13} - \frac{34446781}{412843959000} a^{12} - \frac{3031134173}{412843959000} a^{11} - \frac{1115574689}{206421979500} a^{10} - \frac{22019455}{3302751672} a^{9} + \frac{2511859727}{82568791800} a^{8} + \frac{29002923553}{206421979500} a^{7} + \frac{65562917941}{412843959000} a^{6} - \frac{40536095351}{206421979500} a^{5} + \frac{81986719}{1876563450} a^{4} + \frac{16552546579}{51605494875} a^{3} - \frac{4988304029}{10321098975} a^{2} - \frac{1820351201}{51605494875} a + \frac{1475727119}{9382817250}$

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

Class group and class number

$C_{2}\times C_{6}$, which has order $12$ (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:  \( -\frac{1408897}{5096839000} a^{17} + \frac{31378883}{15290517000} a^{16} - \frac{66212771}{7645258500} a^{15} + \frac{299763671}{15290517000} a^{14} - \frac{980199139}{45871551000} a^{13} - \frac{210458471}{22935775500} a^{12} + \frac{2735295419}{45871551000} a^{11} - \frac{217190789}{5096839000} a^{10} - \frac{381713957}{1529051700} a^{9} + \frac{2662205437}{3058103400} a^{8} - \frac{6797093357}{5096839000} a^{7} + \frac{2620921159}{2548419500} a^{6} + \frac{400444193}{637104875} a^{5} - \frac{2024948501}{417014100} a^{4} + \frac{102313447901}{11467887750} a^{3} - \frac{12581378303}{2293577550} a^{2} - \frac{2619885139}{1911314625} a + \frac{208098308}{173755875} \) (order $6$)
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:  \( 31677601.476 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3$ (as 18T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.11907.1 x3, 3.1.47628.2 x3, 3.1.972.1 x3, 3.1.588.1 x3, 6.0.425329947.3, 6.0.6805279152.4, 6.0.2834352.1, 6.0.1037232.1, 9.1.324121835451456.12 x9

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

Sibling fields

Degree 9 sibling: 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.6.11.8$x^{6} + 12$$6$$1$$11$$S_3$$[5/2]_{2}$
3.6.11.8$x^{6} + 12$$6$$1$$11$$S_3$$[5/2]_{2}$
3.6.11.8$x^{6} + 12$$6$$1$$11$$S_3$$[5/2]_{2}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$