↑

Properties

Label 18.0.12033458364...1875.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{17}\cdot 5^{8}\cdot 13^{4}\cdot 17^{4}$
Root discriminant $19.15$
Ramified primes $3, 5, 13, 17$
Class number $2$
Class group $[2]$
Galois group 18T888

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 3, 7, 8, 13, 34, 15, 9, 32, 10, 17, -4, 15, 6, -2, 2, 4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 4*x^16 + 2*x^15 - 2*x^14 + 6*x^13 + 15*x^12 - 4*x^11 + 17*x^10 + 10*x^9 + 32*x^8 + 9*x^7 + 15*x^6 + 34*x^5 + 13*x^4 + 8*x^3 + 7*x^2 + 3*x + 1)
 
gp: K = bnfinit(x^18 - 2*x^17 + 4*x^16 + 2*x^15 - 2*x^14 + 6*x^13 + 15*x^12 - 4*x^11 + 17*x^10 + 10*x^9 + 32*x^8 + 9*x^7 + 15*x^6 + 34*x^5 + 13*x^4 + 8*x^3 + 7*x^2 + 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 4 x^{16} + 2 x^{15} - 2 x^{14} + 6 x^{13} + 15 x^{12} - 4 x^{11} + 17 x^{10} + 10 x^{9} + 32 x^{8} + 9 x^{7} + 15 x^{6} + 34 x^{5} + 13 x^{4} + 8 x^{3} + 7 x^{2} + 3 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-120334583646716719921875=-\,3^{17}\cdot 5^{8}\cdot 13^{4}\cdot 17^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.15$
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:  $3, 5, 13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{15} a^{13} - \frac{1}{15} a^{12} - \frac{1}{15} a^{11} - \frac{1}{5} a^{10} + \frac{1}{15} a^{8} - \frac{1}{15} a^{7} - \frac{2}{5} a^{6} - \frac{7}{15} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{2}{15} a^{2} + \frac{1}{15} a - \frac{2}{15}$, $\frac{1}{15} a^{14} - \frac{2}{15} a^{12} - \frac{4}{15} a^{11} - \frac{1}{5} a^{10} + \frac{1}{15} a^{9} - \frac{7}{15} a^{7} + \frac{2}{15} a^{6} - \frac{4}{15} a^{5} - \frac{2}{5} a^{4} + \frac{4}{15} a^{3} - \frac{1}{15} a^{2} - \frac{1}{15} a - \frac{2}{15}$, $\frac{1}{15} a^{15} - \frac{1}{15} a^{12} - \frac{1}{3} a^{9} - \frac{2}{5} a^{6} + \frac{2}{5} a^{3} + \frac{1}{15}$, $\frac{1}{45} a^{16} + \frac{1}{45} a^{15} + \frac{1}{45} a^{13} + \frac{7}{45} a^{12} - \frac{7}{45} a^{11} + \frac{14}{45} a^{10} + \frac{1}{3} a^{9} - \frac{1}{15} a^{8} - \frac{8}{45} a^{7} + \frac{17}{45} a^{6} + \frac{11}{45} a^{5} + \frac{7}{45} a^{4} - \frac{22}{45} a^{3} + \frac{7}{15} a^{2} + \frac{2}{5} a + \frac{22}{45}$, $\frac{1}{585} a^{17} - \frac{1}{585} a^{16} + \frac{16}{585} a^{15} - \frac{8}{585} a^{14} - \frac{2}{117} a^{13} + \frac{1}{65} a^{12} - \frac{41}{585} a^{11} + \frac{254}{585} a^{10} + \frac{86}{195} a^{9} + \frac{268}{585} a^{8} + \frac{22}{195} a^{7} - \frac{29}{585} a^{6} + \frac{17}{195} a^{5} + \frac{11}{195} a^{4} - \frac{58}{585} a^{3} + \frac{14}{39} a^{2} - \frac{58}{117} a - \frac{53}{585}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{313}{195} a^{17} - \frac{794}{195} a^{16} + \frac{1706}{195} a^{15} - \frac{346}{195} a^{14} - \frac{322}{195} a^{13} + \frac{692}{65} a^{12} + \frac{1191}{65} a^{11} - \frac{1016}{65} a^{10} + \frac{2439}{65} a^{9} - \frac{889}{195} a^{8} + \frac{3662}{65} a^{7} - \frac{3071}{195} a^{6} + \frac{7292}{195} a^{5} + \frac{6806}{195} a^{4} + \frac{943}{195} a^{3} + \frac{811}{65} a^{2} + \frac{1153}{195} a + \frac{649}{195} \) (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:  \( 19326.0168878 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T888:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 362880
The 36 conjugacy class representatives for t18n888
Character table for t18n888 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 9.5.22253180625.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $18$ R R ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ $18$ R R ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ $18$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $18$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.12.14.4$x^{12} + 6 x^{11} - 6 x^{10} + 6 x^{9} - 3 x^{8} + 9 x^{7} - 6 x^{6} + 9 x^{5} + 9 x^{4} + 9 x^{3} + 9 x^{2} + 9$$6$$2$$14$12T40$[3/2, 3/2]_{2}^{4}$
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.5.0.1$x^{5} - 2 x + 6$$1$$5$$0$$C_5$$[\ ]^{5}$
13.5.0.1$x^{5} - 2 x + 6$$1$$5$$0$$C_5$$[\ ]^{5}$
$17$17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$