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Properties

Label 18.18.3749958646...4000.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{18}\cdot 3^{27}\cdot 5^{3}\cdot 107^{6}$
Root discriminant $64.52$
Ramified primes $2, 3, 5, 107$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T175

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-375, 0, 4500, 0, -19110, 0, 37039, 0, -36303, 0, 18705, 0, -5093, 0, 708, 0, -45, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 45*x^16 + 708*x^14 - 5093*x^12 + 18705*x^10 - 36303*x^8 + 37039*x^6 - 19110*x^4 + 4500*x^2 - 375)
 
gp: K = bnfinit(x^18 - 45*x^16 + 708*x^14 - 5093*x^12 + 18705*x^10 - 36303*x^8 + 37039*x^6 - 19110*x^4 + 4500*x^2 - 375, 1)
 

Normalized defining polynomial

\( x^{18} - 45 x^{16} + 708 x^{14} - 5093 x^{12} + 18705 x^{10} - 36303 x^{8} + 37039 x^{6} - 19110 x^{4} + 4500 x^{2} - 375 \)

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:  \(374995864672776683690097475584000=2^{18}\cdot 3^{27}\cdot 5^{3}\cdot 107^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.52$
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:  $2, 3, 5, 107$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{10} - \frac{1}{6} a^{8} + \frac{1}{6} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{11} - \frac{1}{6} a^{9} + \frac{1}{6} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{90} a^{14} + \frac{1}{18} a^{12} + \frac{1}{30} a^{10} - \frac{43}{90} a^{8} - \frac{7}{18} a^{6} - \frac{1}{30} a^{4} - \frac{7}{30} a^{2} + \frac{1}{3}$, $\frac{1}{90} a^{15} + \frac{1}{18} a^{13} + \frac{1}{30} a^{11} - \frac{43}{90} a^{9} - \frac{7}{18} a^{7} - \frac{1}{30} a^{5} - \frac{7}{30} a^{3} + \frac{1}{3} a$, $\frac{1}{72528541650} a^{16} - \frac{45030119}{14505708330} a^{14} - \frac{94687489}{24176180550} a^{12} + \frac{254492207}{72528541650} a^{10} + \frac{2125225901}{14505708330} a^{8} + \frac{2071025083}{8058726850} a^{6} + \frac{3499296263}{24176180550} a^{4} + \frac{1170756958}{2417618055} a^{2} - \frac{59204031}{161174537}$, $\frac{1}{72528541650} a^{17} - \frac{45030119}{14505708330} a^{15} - \frac{94687489}{24176180550} a^{13} + \frac{254492207}{72528541650} a^{11} + \frac{2125225901}{14505708330} a^{9} + \frac{2071025083}{8058726850} a^{7} + \frac{3499296263}{24176180550} a^{5} + \frac{1170756958}{2417618055} a^{3} - \frac{59204031}{161174537} a$

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:  \( 101858915718 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T175:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 576
The 28 conjugacy class representatives for t18n175
Character table for t18n175 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.3.321.1, 9.9.1953114230889.1

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

Sibling fields

Degree 12 sibling: data not computed
Degree 16 sibling: data not computed
Degree 18 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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.6.6.6$x^{6} - 13 x^{4} + 7 x^{2} - 3$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
$3$3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
$5$5.6.3.2$x^{6} - 25 x^{2} + 250$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
5.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$107$107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
107.4.2.1$x^{4} + 963 x^{2} + 286225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$