↑

Properties

Label 18.0.83769291871...8432.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{25}\cdot 17^{6}$
Root discriminant $18.77$
Ramified primes $2, 3, 17$
Class number $3$
Class group $[3]$
Galois group $S_3^2$ (as 18T11)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, -96, 96, -120, 144, -198, 207, -183, 147, -110, 123, -105, 102, -45, 9, 0, 3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 3*x^16 + 9*x^14 - 45*x^13 + 102*x^12 - 105*x^11 + 123*x^10 - 110*x^9 + 147*x^8 - 183*x^7 + 207*x^6 - 198*x^5 + 144*x^4 - 120*x^3 + 96*x^2 - 96*x + 64)
 
gp: K = bnfinit(x^18 - 3*x^17 + 3*x^16 + 9*x^14 - 45*x^13 + 102*x^12 - 105*x^11 + 123*x^10 - 110*x^9 + 147*x^8 - 183*x^7 + 207*x^6 - 198*x^5 + 144*x^4 - 120*x^3 + 96*x^2 - 96*x + 64, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 3 x^{16} + 9 x^{14} - 45 x^{13} + 102 x^{12} - 105 x^{11} + 123 x^{10} - 110 x^{9} + 147 x^{8} - 183 x^{7} + 207 x^{6} - 198 x^{5} + 144 x^{4} - 120 x^{3} + 96 x^{2} - 96 x + 64 \)

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:  \(-83769291871618924818432=-\,2^{12}\cdot 3^{25}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.77$
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:  $2, 3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3}$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{6} a$, $\frac{1}{18} a^{11} + \frac{1}{18} a^{10} - \frac{1}{9} a^{9} - \frac{1}{2} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} + \frac{1}{3} a^{3} - \frac{1}{18} a^{2} + \frac{5}{18} a + \frac{1}{9}$, $\frac{1}{36} a^{12} - \frac{1}{36} a^{11} + \frac{1}{18} a^{10} - \frac{5}{36} a^{9} + \frac{5}{12} a^{8} - \frac{1}{3} a^{7} - \frac{1}{12} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{5}{36} a^{3} - \frac{11}{36} a^{2} + \frac{1}{9} a - \frac{1}{9}$, $\frac{1}{36} a^{13} - \frac{1}{36} a^{11} + \frac{1}{36} a^{10} + \frac{1}{18} a^{9} - \frac{5}{12} a^{8} + \frac{1}{4} a^{7} - \frac{1}{3} a^{6} + \frac{5}{12} a^{5} - \frac{1}{36} a^{4} - \frac{1}{2} a^{3} - \frac{5}{36} a^{2} - \frac{4}{9} a + \frac{1}{9}$, $\frac{1}{36} a^{14} - \frac{1}{18} a^{10} + \frac{1}{9} a^{9} - \frac{1}{3} a^{8} - \frac{1}{6} a^{7} + \frac{1}{3} a^{6} - \frac{5}{18} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{2} + \frac{7}{18} a + \frac{2}{9}$, $\frac{1}{72} a^{15} - \frac{1}{72} a^{14} - \frac{1}{72} a^{13} + \frac{1}{72} a^{11} - \frac{5}{72} a^{10} - \frac{5}{36} a^{9} - \frac{11}{24} a^{8} + \frac{1}{24} a^{7} + \frac{13}{36} a^{6} + \frac{7}{72} a^{5} + \frac{25}{72} a^{4} - \frac{11}{24} a^{3} - \frac{7}{18} a^{2} - \frac{1}{18} a + \frac{2}{9}$, $\frac{1}{18288} a^{16} + \frac{1}{18288} a^{15} - \frac{1}{2032} a^{14} - \frac{13}{4572} a^{13} - \frac{55}{18288} a^{12} - \frac{449}{18288} a^{11} + \frac{101}{9144} a^{10} - \frac{2273}{18288} a^{9} + \frac{1015}{2032} a^{8} - \frac{2717}{9144} a^{7} + \frac{9035}{18288} a^{6} + \frac{1343}{6096} a^{5} + \frac{8923}{18288} a^{4} - \frac{2293}{9144} a^{3} - \frac{472}{1143} a^{2} + \frac{341}{1143} a + \frac{32}{1143}$, $\frac{1}{621755424} a^{17} + \frac{539}{69083936} a^{16} - \frac{3107675}{621755424} a^{15} + \frac{401059}{310877712} a^{14} + \frac{5605493}{621755424} a^{13} - \frac{5656063}{621755424} a^{12} + \frac{119305}{17270984} a^{11} + \frac{18074167}{621755424} a^{10} + \frac{11433911}{207251808} a^{9} - \frac{1254101}{77719428} a^{8} - \frac{4749959}{207251808} a^{7} + \frac{24370427}{621755424} a^{6} - \frac{129276647}{621755424} a^{5} - \frac{26145361}{77719428} a^{4} + \frac{2118883}{38859714} a^{3} + \frac{426889}{8635492} a^{2} - \frac{11450359}{38859714} a + \frac{332614}{6476619}$

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:  \( \frac{317}{305982} a^{17} - \frac{22139}{2447856} a^{16} + \frac{29579}{2447856} a^{15} - \frac{14087}{2447856} a^{14} + \frac{24251}{1223928} a^{13} - \frac{308551}{2447856} a^{12} + \frac{656989}{2447856} a^{11} - \frac{259007}{611964} a^{10} + \frac{1045139}{2447856} a^{9} - \frac{2786671}{2447856} a^{8} + \frac{171247}{305982} a^{7} - \frac{2224289}{2447856} a^{6} + \frac{282023}{2447856} a^{5} - \frac{2276851}{2447856} a^{4} + \frac{396553}{611964} a^{3} - \frac{155285}{305982} a^{2} + \frac{91346}{152991} a + \frac{78346}{152991} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 23410.949000205437 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.108.1 x3, 3.1.459.1, 6.0.632043.1, 6.0.34992.1, 9.1.167102056512.1 x3

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 6 sibling: data not computed
Degree 9 sibling: data not computed
Degree 12 sibling: data not computed
Degree 18 siblings: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$$[\ ]_{3}^{2}$
3Data not computed
$17$17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$