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Properties

Label 18.0.24930271280...5063.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{12}\cdot 23^{9}$
Root discriminant $17.55$
Ramified primes $7, 23$
Class number $3$
Class group $[3]$
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 21, -43, 105, -126, 124, -100, 262, -260, 286, -162, 110, -28, 35, -8, 7, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 7*x^16 - 8*x^15 + 35*x^14 - 28*x^13 + 110*x^12 - 162*x^11 + 286*x^10 - 260*x^9 + 262*x^8 - 100*x^7 + 124*x^6 - 126*x^5 + 105*x^4 - 43*x^3 + 21*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^18 - 3*x^17 + 7*x^16 - 8*x^15 + 35*x^14 - 28*x^13 + 110*x^12 - 162*x^11 + 286*x^10 - 260*x^9 + 262*x^8 - 100*x^7 + 124*x^6 - 126*x^5 + 105*x^4 - 43*x^3 + 21*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 7 x^{16} - 8 x^{15} + 35 x^{14} - 28 x^{13} + 110 x^{12} - 162 x^{11} + 286 x^{10} - 260 x^{9} + 262 x^{8} - 100 x^{7} + 124 x^{6} - 126 x^{5} + 105 x^{4} - 43 x^{3} + 21 x^{2} - 6 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-24930271280154907835063=-\,7^{12}\cdot 23^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.55$
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:  $7, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{2} - \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{9} - \frac{1}{6} a^{8} + \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{6} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{6} a^{13} - \frac{1}{2} a^{6} + \frac{1}{3}$, $\frac{1}{36} a^{14} + \frac{1}{18} a^{12} + \frac{1}{18} a^{11} - \frac{1}{6} a^{9} + \frac{2}{9} a^{8} - \frac{2}{9} a^{7} - \frac{1}{3} a^{6} - \frac{1}{18} a^{5} - \frac{1}{6} a^{4} - \frac{1}{9} a^{3} + \frac{7}{18} a^{2} + \frac{4}{9} a - \frac{5}{36}$, $\frac{1}{36} a^{15} + \frac{1}{18} a^{13} + \frac{1}{18} a^{12} + \frac{1}{18} a^{9} - \frac{2}{9} a^{8} - \frac{1}{18} a^{6} - \frac{1}{2} a^{5} - \frac{4}{9} a^{4} - \frac{4}{9} a^{3} + \frac{5}{18} a^{2} - \frac{17}{36} a - \frac{1}{3}$, $\frac{1}{3516948} a^{16} - \frac{28103}{3516948} a^{15} + \frac{7505}{1758474} a^{14} - \frac{111379}{1758474} a^{13} - \frac{11551}{195386} a^{12} + \frac{69065}{1758474} a^{11} - \frac{2933}{1758474} a^{10} - \frac{14417}{195386} a^{9} + \frac{346771}{1758474} a^{8} + \frac{33217}{586158} a^{7} - \frac{35005}{1758474} a^{6} + \frac{292529}{1758474} a^{5} - \frac{762421}{1758474} a^{4} + \frac{29087}{1758474} a^{3} - \frac{139589}{1172316} a^{2} + \frac{86671}{1172316} a + \frac{134171}{879237}$, $\frac{1}{3516948} a^{17} - \frac{13387}{3516948} a^{15} - \frac{9929}{879237} a^{14} + \frac{15185}{293079} a^{13} - \frac{39620}{879237} a^{12} - \frac{5627}{293079} a^{11} + \frac{46213}{879237} a^{10} - \frac{607}{293079} a^{9} - \frac{155537}{1758474} a^{8} + \frac{282589}{1758474} a^{7} + \frac{255455}{879237} a^{6} + \frac{340909}{879237} a^{5} - \frac{129758}{879237} a^{4} + \frac{1408967}{3516948} a^{3} - \frac{62552}{879237} a^{2} - \frac{1122079}{3516948} a - \frac{391295}{1758474}$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2564.16185262 \)
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

\(\Q(\sqrt{-23}) \), 3.1.23.1 x3, \(\Q(\zeta_{7})^+\), 6.0.12167.1, 6.0.29212967.2 x2, 6.0.29212967.1, 9.3.1431435383.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.29212967.2
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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$23$23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$