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Properties

Label 18.0.18294428062...7584.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 7^{12}\cdot 19^{9}$
Root discriminant $25.32$
Ramified primes $2, 7, 19$
Class number $3$
Class group $[3]$
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![2736, -20064, 70784, -154816, 231552, -249352, 201816, -130472, 73892, -39456, 19412, -8326, 3191, -1156, 383, -114, 33, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 + 33*x^16 - 114*x^15 + 383*x^14 - 1156*x^13 + 3191*x^12 - 8326*x^11 + 19412*x^10 - 39456*x^9 + 73892*x^8 - 130472*x^7 + 201816*x^6 - 249352*x^5 + 231552*x^4 - 154816*x^3 + 70784*x^2 - 20064*x + 2736)
 
gp: K = bnfinit(x^18 - 8*x^17 + 33*x^16 - 114*x^15 + 383*x^14 - 1156*x^13 + 3191*x^12 - 8326*x^11 + 19412*x^10 - 39456*x^9 + 73892*x^8 - 130472*x^7 + 201816*x^6 - 249352*x^5 + 231552*x^4 - 154816*x^3 + 70784*x^2 - 20064*x + 2736, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} + 33 x^{16} - 114 x^{15} + 383 x^{14} - 1156 x^{13} + 3191 x^{12} - 8326 x^{11} + 19412 x^{10} - 39456 x^{9} + 73892 x^{8} - 130472 x^{7} + 201816 x^{6} - 249352 x^{5} + 231552 x^{4} - 154816 x^{3} + 70784 x^{2} - 20064 x + 2736 \)

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:  \(-18294428062468623673667584=-\,2^{12}\cdot 7^{12}\cdot 19^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.32$
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:  $2, 7, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{12} a^{9} - \frac{1}{12} a^{8} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{5}{12} a^{5} + \frac{5}{12} a^{4} + \frac{1}{3} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{10} - \frac{1}{12} a^{8} - \frac{1}{6} a^{7} - \frac{1}{12} a^{6} - \frac{1}{6} a^{5} + \frac{5}{12} a^{4} - \frac{1}{6} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{8} + \frac{1}{12} a^{7} + \frac{1}{6} a^{6} - \frac{1}{6} a^{5} - \frac{1}{12} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{6} a$, $\frac{1}{24} a^{12} - \frac{1}{24} a^{10} - \frac{1}{24} a^{8} + \frac{1}{6} a^{7} + \frac{5}{24} a^{6} - \frac{1}{3} a^{5} + \frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{24} a^{13} - \frac{1}{24} a^{11} - \frac{1}{24} a^{9} - \frac{1}{8} a^{7} - \frac{1}{6} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} + \frac{1}{6} a$, $\frac{1}{144} a^{14} + \frac{1}{72} a^{13} - \frac{1}{48} a^{12} + \frac{1}{36} a^{11} + \frac{1}{48} a^{10} - \frac{1}{72} a^{9} + \frac{11}{144} a^{8} + \frac{1}{36} a^{7} + \frac{7}{36} a^{6} - \frac{7}{36} a^{5} - \frac{13}{36} a^{4} - \frac{1}{36} a^{2}$, $\frac{1}{1440} a^{15} + \frac{1}{360} a^{14} + \frac{5}{288} a^{13} + \frac{1}{90} a^{12} - \frac{49}{1440} a^{11} - \frac{5}{144} a^{10} - \frac{41}{1440} a^{9} - \frac{2}{45} a^{8} - \frac{103}{720} a^{6} - \frac{23}{120} a^{5} + \frac{7}{90} a^{4} + \frac{113}{360} a^{3} + \frac{31}{90} a^{2} + \frac{1}{3} a + \frac{3}{20}$, $\frac{1}{4320} a^{16} - \frac{11}{4320} a^{14} - \frac{1}{1080} a^{13} + \frac{7}{4320} a^{12} + \frac{11}{720} a^{11} - \frac{3}{160} a^{10} + \frac{1}{216} a^{9} - \frac{11}{180} a^{8} + \frac{457}{2160} a^{7} - \frac{43}{180} a^{6} + \frac{11}{45} a^{5} - \frac{17}{40} a^{4} + \frac{83}{270} a^{3} + \frac{53}{135} a^{2} + \frac{29}{180} a - \frac{1}{30}$, $\frac{1}{5841255867840} a^{17} - \frac{61264333}{584125586784} a^{16} + \frac{666851251}{5841255867840} a^{15} - \frac{3689448223}{2920627933920} a^{14} - \frac{116236643443}{5841255867840} a^{13} + \frac{2648915627}{1460313966960} a^{12} + \frac{54238309007}{1947085289280} a^{11} + \frac{5409473597}{584125586784} a^{10} + \frac{60853489577}{2920627933920} a^{9} - \frac{211857668867}{2920627933920} a^{8} - \frac{14143562783}{1460313966960} a^{7} + \frac{90660966097}{486771322320} a^{6} - \frac{53075472473}{162257107440} a^{5} - \frac{30236583101}{730156983480} a^{4} + \frac{71699295749}{146031396696} a^{3} + \frac{268725277531}{730156983480} a^{2} + \frac{596899219}{3202442910} a - \frac{490221533}{2134961940}$

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

Class group and class number

$C_{3}$, which has order $3$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 229812.209675 \)
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

\(\Q(\sqrt{-19}) \), 3.1.3724.2 x3, 3.1.76.1 x3, 3.1.931.1 x3, 3.1.3724.1 x3, 6.0.263495344.2, 6.0.109744.2, 6.0.16468459.2, 6.0.263495344.1, 9.1.981256661056.1 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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$$[\ ]_{3}^{2}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$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$$[\ ]_{3}^{3}$
$19$19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$