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Properties

Label 18.6.54257845827...0000.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 5^{9}\cdot 7^{14}$
Root discriminant $16.12$
Ramified primes $2, 5, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 11, -25, -33, 135, -51, -123, 177, -84, -70, 135, -117, 71, -19, -6, 9, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 + 9*x^16 - 6*x^15 - 19*x^14 + 71*x^13 - 117*x^12 + 135*x^11 - 70*x^10 - 84*x^9 + 177*x^8 - 123*x^7 - 51*x^6 + 135*x^5 - 33*x^4 - 25*x^3 + 11*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^18 - 5*x^17 + 9*x^16 - 6*x^15 - 19*x^14 + 71*x^13 - 117*x^12 + 135*x^11 - 70*x^10 - 84*x^9 + 177*x^8 - 123*x^7 - 51*x^6 + 135*x^5 - 33*x^4 - 25*x^3 + 11*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} + 9 x^{16} - 6 x^{15} - 19 x^{14} + 71 x^{13} - 117 x^{12} + 135 x^{11} - 70 x^{10} - 84 x^{9} + 177 x^{8} - 123 x^{7} - 51 x^{6} + 135 x^{5} - 33 x^{4} - 25 x^{3} + 11 x^{2} - 8 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5425784582792000000000=2^{12}\cdot 5^{9}\cdot 7^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.12$
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:  $2, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{71} a^{16} + \frac{8}{71} a^{15} - \frac{18}{71} a^{14} - \frac{10}{71} a^{13} + \frac{8}{71} a^{12} - \frac{6}{71} a^{11} + \frac{35}{71} a^{10} + \frac{27}{71} a^{9} + \frac{27}{71} a^{8} - \frac{4}{71} a^{7} - \frac{4}{71} a^{6} - \frac{6}{71} a^{5} - \frac{31}{71} a^{4} + \frac{21}{71} a^{3} - \frac{30}{71} a^{2} + \frac{29}{71} a - \frac{13}{71}$, $\frac{1}{22182208010101} a^{17} + \frac{151664709536}{22182208010101} a^{16} + \frac{9650096508899}{22182208010101} a^{15} - \frac{9676953003084}{22182208010101} a^{14} - \frac{10411667037479}{22182208010101} a^{13} - \frac{4793278724366}{22182208010101} a^{12} + \frac{3602201010574}{22182208010101} a^{11} + \frac{4773758414474}{22182208010101} a^{10} + \frac{3162781026185}{22182208010101} a^{9} - \frac{6561253702937}{22182208010101} a^{8} - \frac{4992331982919}{22182208010101} a^{7} + \frac{1003279965773}{22182208010101} a^{6} + \frac{9467454354780}{22182208010101} a^{5} + \frac{2822276462171}{22182208010101} a^{4} - \frac{10939288680720}{22182208010101} a^{3} - \frac{1820685470348}{22182208010101} a^{2} - \frac{2902349174430}{22182208010101} a + \frac{4898940833974}{22182208010101}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 6358.139706194206 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), 3.1.140.1, 6.2.98000.1, 6.6.300125.1, 9.3.6588344000.1

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 12 sibling: 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
2Data not computed
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$