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Properties

Label 18.0.15485076856...2467.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{21}\cdot 23^{6}$
Root discriminant $10.25$
Ramified primes $3, 23$
Class number $1$
Class group Trivial
Galois group $C_3.S_3^2$ (as 18T57)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 21, -67, 147, -243, 334, -378, 378, -335, 252, -189, 109, -63, 36, -9, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 9*x^16 - 9*x^15 + 36*x^14 - 63*x^13 + 109*x^12 - 189*x^11 + 252*x^10 - 335*x^9 + 378*x^8 - 378*x^7 + 334*x^6 - 243*x^5 + 147*x^4 - 67*x^3 + 21*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^18 + 9*x^16 - 9*x^15 + 36*x^14 - 63*x^13 + 109*x^12 - 189*x^11 + 252*x^10 - 335*x^9 + 378*x^8 - 378*x^7 + 334*x^6 - 243*x^5 + 147*x^4 - 67*x^3 + 21*x^2 - 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} + 9 x^{16} - 9 x^{15} + 36 x^{14} - 63 x^{13} + 109 x^{12} - 189 x^{11} + 252 x^{10} - 335 x^{9} + 378 x^{8} - 378 x^{7} + 334 x^{6} - 243 x^{5} + 147 x^{4} - 67 x^{3} + 21 x^{2} - 3 x + 1 \)

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:  \(-1548507685660102467=-\,3^{21}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $10.25$
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:  $3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{1153182335} a^{17} - \frac{394336921}{1153182335} a^{16} - \frac{85928896}{230636467} a^{15} - \frac{97238009}{1153182335} a^{14} - \frac{49031889}{230636467} a^{13} - \frac{138175543}{1153182335} a^{12} + \frac{462712092}{1153182335} a^{11} - \frac{279666761}{1153182335} a^{10} - \frac{280014797}{1153182335} a^{9} - \frac{524613583}{1153182335} a^{8} + \frac{63869246}{1153182335} a^{7} - \frac{17051817}{67834255} a^{6} - \frac{77106787}{1153182335} a^{5} - \frac{214113886}{1153182335} a^{4} + \frac{560320823}{1153182335} a^{3} - \frac{96385234}{230636467} a^{2} - \frac{142185499}{1153182335} a + \frac{564604746}{1153182335}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( -\frac{3077572}{11195945} a^{17} - \frac{313528}{11195945} a^{16} - \frac{5796705}{2239189} a^{15} + \frac{22523788}{11195945} a^{14} - \frac{24327282}{2239189} a^{13} + \frac{172258676}{11195945} a^{12} - \frac{359270994}{11195945} a^{11} + \frac{555197617}{11195945} a^{10} - \frac{776133451}{11195945} a^{9} + \frac{1033752351}{11195945} a^{8} - \frac{1120967627}{11195945} a^{7} + \frac{68522744}{658585} a^{6} - \frac{989519121}{11195945} a^{5} + \frac{715640877}{11195945} a^{4} - \frac{427450321}{11195945} a^{3} + \frac{36791096}{2239189} a^{2} - \frac{55256652}{11195945} a + \frac{15041843}{11195945} \) (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:  \( 78.7017437829 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3.S_3^2$ (as 18T57):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 11 conjugacy class representatives for $C_3.S_3^2$
Character table for $C_3.S_3^2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.23.1, 6.0.14283.1, 9.1.239483061.1 x3

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

Sibling fields

Degree 9 siblings: 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
3Data not computed
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$