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Properties

Label 18.0.35347840065...1875.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{21}\cdot 5^{12}\cdot 7^{12}$
Root discriminant $38.55$
Ramified primes $3, 5, 7$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![48, -432, 1740, -3996, 5559, -4557, 1849, 192, -1044, 1224, -654, -42, 358, -336, 216, -108, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 216*x^14 - 336*x^13 + 358*x^12 - 42*x^11 - 654*x^10 + 1224*x^9 - 1044*x^8 + 192*x^7 + 1849*x^6 - 4557*x^5 + 5559*x^4 - 3996*x^3 + 1740*x^2 - 432*x + 48)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 216*x^14 - 336*x^13 + 358*x^12 - 42*x^11 - 654*x^10 + 1224*x^9 - 1044*x^8 + 192*x^7 + 1849*x^6 - 4557*x^5 + 5559*x^4 - 3996*x^3 + 1740*x^2 - 432*x + 48, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 108 x^{15} + 216 x^{14} - 336 x^{13} + 358 x^{12} - 42 x^{11} - 654 x^{10} + 1224 x^{9} - 1044 x^{8} + 192 x^{7} + 1849 x^{6} - 4557 x^{5} + 5559 x^{4} - 3996 x^{3} + 1740 x^{2} - 432 x + 48 \)

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:  \(-35347840065093568067138671875=-\,3^{21}\cdot 5^{12}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.55$
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:  $3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{8} a^{5} - \frac{1}{2} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{6} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} + \frac{1}{8} a^{7} + \frac{1}{8} a^{3} - \frac{3}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} + \frac{1}{16} a^{8} + \frac{1}{16} a^{7} - \frac{3}{16} a^{6} - \frac{3}{16} a^{5} - \frac{3}{16} a^{4} - \frac{3}{16} a^{3} - \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{80} a^{15} + \frac{1}{40} a^{13} + \frac{1}{40} a^{12} + \frac{1}{20} a^{10} - \frac{1}{40} a^{9} + \frac{1}{40} a^{8} - \frac{7}{40} a^{7} - \frac{1}{8} a^{6} + \frac{1}{10} a^{5} - \frac{1}{20} a^{4} + \frac{17}{80} a^{3} + \frac{3}{8} a^{2} - \frac{7}{20} a + \frac{1}{5}$, $\frac{1}{195040} a^{16} - \frac{1}{24380} a^{15} - \frac{51}{24380} a^{14} + \frac{749}{48760} a^{13} + \frac{729}{12190} a^{12} - \frac{16}{265} a^{11} - \frac{6057}{97520} a^{10} + \frac{833}{12190} a^{9} + \frac{397}{4876} a^{8} - \frac{1767}{48760} a^{7} - \frac{211}{6095} a^{6} + \frac{1}{6095} a^{5} - \frac{8351}{195040} a^{4} + \frac{251}{1060} a^{3} - \frac{2293}{12190} a^{2} - \frac{173}{4876} a + \frac{1076}{6095}$, $\frac{1}{9166880} a^{17} + \frac{3}{1833376} a^{16} + \frac{17989}{4583440} a^{15} + \frac{2901}{4583440} a^{14} - \frac{26759}{4583440} a^{13} + \frac{10081}{199280} a^{12} + \frac{54531}{1145860} a^{11} - \frac{1163}{1145860} a^{10} - \frac{478763}{4583440} a^{9} + \frac{160801}{4583440} a^{8} + \frac{177427}{4583440} a^{7} - \frac{303147}{4583440} a^{6} - \frac{194125}{1833376} a^{5} - \frac{91673}{398560} a^{4} + \frac{871981}{2291720} a^{3} - \frac{267541}{2291720} a^{2} - \frac{20659}{286465} a + \frac{4007}{24910}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( -\frac{156909}{1145860} a^{17} + \frac{2667453}{2291720} a^{16} - \frac{1091333}{229172} a^{15} + \frac{5700183}{458344} a^{14} - \frac{2338609}{99640} a^{13} + \frac{7918365}{229172} a^{12} - \frac{73908587}{2291720} a^{11} - \frac{22136763}{2291720} a^{10} + \frac{96416343}{1145860} a^{9} - \frac{57733845}{458344} a^{8} + \frac{187720307}{2291720} a^{7} + \frac{13895553}{1145860} a^{6} - \frac{112686275}{458344} a^{5} + \frac{575120241}{1145860} a^{4} - \frac{147992618}{286465} a^{3} + \frac{342958449}{1145860} a^{2} - \frac{5625996}{57293} a + \frac{4348757}{286465} \) (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:  \( 161495309.756 \) (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

\(\Q(\sqrt{-3}) \), 3.1.33075.1 x3, 3.1.1323.1 x3, 3.1.675.1 x3, 3.1.3675.1 x3, 6.0.3281866875.1, 6.0.5250987.1, 6.0.1366875.1, 6.0.40516875.1, 9.1.108547746890625.3 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{9}$ R R 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
$3$3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{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}$