↑

Properties

Label 18.18.1482416151...0000.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{12}\cdot 3^{32}\cdot 5^{9}$
Root discriminant $25.03$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, -21, 162, -12, -1077, 1038, 2118, -2604, -1987, 2523, 1122, -1143, -381, 237, 57, -24, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 24*x^16 + 57*x^15 + 237*x^14 - 381*x^13 - 1143*x^12 + 1122*x^11 + 2523*x^10 - 1987*x^9 - 2604*x^8 + 2118*x^7 + 1038*x^6 - 1077*x^5 - 12*x^4 + 162*x^3 - 21*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^18 - 3*x^17 - 24*x^16 + 57*x^15 + 237*x^14 - 381*x^13 - 1143*x^12 + 1122*x^11 + 2523*x^10 - 1987*x^9 - 2604*x^8 + 2118*x^7 + 1038*x^6 - 1077*x^5 - 12*x^4 + 162*x^3 - 21*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 24 x^{16} + 57 x^{15} + 237 x^{14} - 381 x^{13} - 1143 x^{12} + 1122 x^{11} + 2523 x^{10} - 1987 x^{9} - 2604 x^{8} + 2118 x^{7} + 1038 x^{6} - 1077 x^{5} - 12 x^{4} + 162 x^{3} - 21 x^{2} - 6 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(14824161510814728000000000=2^{12}\cdot 3^{32}\cdot 5^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.03$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{6}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{7}$, $\frac{1}{5055207} a^{17} + \frac{668482}{5055207} a^{16} + \frac{1360}{5055207} a^{15} - \frac{797603}{5055207} a^{14} + \frac{521624}{5055207} a^{13} - \frac{249187}{5055207} a^{12} - \frac{92281}{1685069} a^{11} + \frac{227110}{1685069} a^{10} - \frac{32502}{1685069} a^{9} + \frac{2223730}{5055207} a^{8} + \frac{1031233}{5055207} a^{7} + \frac{381154}{5055207} a^{6} - \frac{1865693}{5055207} a^{5} + \frac{868547}{5055207} a^{4} - \frac{103495}{5055207} a^{3} - \frac{325660}{1685069} a^{2} + \frac{294210}{1685069} a + \frac{458444}{1685069}$

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:  $17$
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:  \( 4070482.18025 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{5}) \), 3.3.1620.1 x3, \(\Q(\zeta_{9})^+\), 6.6.13122000.1, 6.6.13122000.2 x2, 6.6.820125.1, 9.9.344373768000.1 x3

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

Sibling fields

Degree 6 sibling: 6.6.13122000.2
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 R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
3Data not computed
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$