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Properties

Label 18.0.14070432741...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{20}\cdot 5^{12}\cdot 7^{9}$
Root discriminant $41.62$
Ramified primes $2, 3, 5, 7$
Class number $18$ (GRH)
Class group $[3, 6]$ (GRH)
Galois group $C_2\times C_3:S_3$ (as 18T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![512, 0, 0, 7360, 0, 0, 33352, 0, 0, -6435, 0, 0, 599, 0, 0, -35, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 35*x^15 + 599*x^12 - 6435*x^9 + 33352*x^6 + 7360*x^3 + 512)
 
gp: K = bnfinit(x^18 - 35*x^15 + 599*x^12 - 6435*x^9 + 33352*x^6 + 7360*x^3 + 512, 1)
 

Normalized defining polynomial

\( x^{18} - 35 x^{15} + 599 x^{12} - 6435 x^{9} + 33352 x^{6} + 7360 x^{3} + 512 \)

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:  \(-140704327411684407000000000000=-\,2^{12}\cdot 3^{20}\cdot 5^{12}\cdot 7^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.62$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{4} a^{11} + \frac{1}{4} a^{8} - \frac{1}{4} a^{5} + \frac{1}{4} a^{2}$, $\frac{1}{72} a^{12} - \frac{3}{8} a^{9} - \frac{1}{8} a^{6} - \frac{3}{8} a^{3} + \frac{2}{9}$, $\frac{1}{144} a^{13} - \frac{3}{16} a^{10} + \frac{7}{16} a^{7} - \frac{3}{16} a^{4} + \frac{1}{9} a$, $\frac{1}{288} a^{14} - \frac{3}{32} a^{11} + \frac{7}{32} a^{8} + \frac{13}{32} a^{5} + \frac{1}{18} a^{2}$, $\frac{1}{3772510272} a^{15} - \frac{11787011}{3772510272} a^{12} + \frac{144812351}{419167808} a^{9} - \frac{187267371}{419167808} a^{6} + \frac{143282045}{471563784} a^{3} - \frac{23585114}{58945473}$, $\frac{1}{22635061632} a^{16} + \frac{1}{11317530816} a^{15} + \frac{1}{864} a^{14} - \frac{11787011}{22635061632} a^{13} + \frac{40608965}{11317530816} a^{12} + \frac{5}{96} a^{11} + \frac{563980159}{2515006848} a^{10} - \frac{143847795}{419167808} a^{9} + \frac{47}{96} a^{8} - \frac{606435179}{2515006848} a^{7} + \frac{199557423}{419167808} a^{6} + \frac{5}{96} a^{5} - \frac{799845523}{2829382704} a^{4} - \frac{252559079}{707345676} a^{3} + \frac{11}{108} a^{2} + \frac{153251305}{353672838} a + \frac{48459353}{176836419}$, $\frac{1}{45270123264} a^{17} + \frac{1}{11317530816} a^{15} - \frac{7131443}{5030013696} a^{14} + \frac{1}{432} a^{13} - \frac{11787011}{11317530816} a^{12} + \frac{302000279}{5030013696} a^{11} + \frac{5}{48} a^{10} + \frac{563980159}{1257503424} a^{9} + \frac{1960967645}{5030013696} a^{8} - \frac{1}{48} a^{7} - \frac{606435179}{1257503424} a^{6} - \frac{136821611}{707345676} a^{5} + \frac{5}{48} a^{4} + \frac{614845829}{1414691352} a^{3} + \frac{4511491}{39296982} a^{2} + \frac{11}{54} a - \frac{23585114}{176836419}$

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

Class group and class number

$C_{3}\times C_{6}$, which has order $18$ (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:  \( 5863974.111036323 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3:S_3$ (as 18T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 12 conjugacy class representatives for $C_2\times C_3:S_3$
Character table for $C_2\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{-7}) \), 3.1.108.1, 3.1.300.1, 3.1.675.1, 3.1.2700.1, 6.0.4000752.4, 6.0.156279375.1, 6.0.30870000.1, 6.0.2500470000.2, 9.1.59049000000.1

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

Sibling fields

Galois closure: data not computed
Degree 18 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 R R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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}$
2.6.4.1$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$3.6.6.3$x^{6} + 3 x^{4} + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
3.12.14.6$x^{12} + 3 x^{11} + 3 x^{10} - 6 x^{9} + 3 x^{8} + 9 x^{7} + 9 x^{4} + 9 x^{3} + 9$$6$$2$$14$$D_6$$[3/2]_{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$$[\ ]_{3}^{2}$
$7$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$