↑

Properties

Label 18.2.20718915703...5024.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{12}\cdot 3^{20}\cdot 29^{9}$
Root discriminant $28.97$
Ramified primes $2, 3, 29$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $S_3^2$ (as 18T9)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, 0, 0, 896, 0, 0, -1200, 0, 0, 288, 0, 0, 96, 0, 0, -20, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 20*x^15 + 96*x^12 + 288*x^9 - 1200*x^6 + 896*x^3 + 64)
 
gp: K = bnfinit(x^18 - 20*x^15 + 96*x^12 + 288*x^9 - 1200*x^6 + 896*x^3 + 64, 1)
 

Normalized defining polynomial

\( x^{18} - 20 x^{15} + 96 x^{12} + 288 x^{9} - 1200 x^{6} + 896 x^{3} + 64 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(207189157034762041833345024=2^{12}\cdot 3^{20}\cdot 29^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.97$
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:  $2, 3, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{4} a^{6}$, $\frac{1}{4} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{16} a^{9} - \frac{1}{4} a^{3} - \frac{1}{2}$, $\frac{1}{16} a^{10} - \frac{1}{4} a^{4} - \frac{1}{2} a$, $\frac{1}{16} a^{11} - \frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{480} a^{12} + \frac{1}{240} a^{9} - \frac{3}{40} a^{6} - \frac{1}{30} a^{3} - \frac{1}{30}$, $\frac{1}{480} a^{13} + \frac{1}{240} a^{10} - \frac{3}{40} a^{7} - \frac{1}{30} a^{4} - \frac{1}{30} a$, $\frac{1}{480} a^{14} + \frac{1}{240} a^{11} - \frac{3}{40} a^{8} - \frac{1}{30} a^{5} - \frac{1}{30} a^{2}$, $\frac{1}{4800} a^{15} - \frac{1}{1200} a^{12} + \frac{11}{400} a^{9} + \frac{1}{24} a^{6} + \frac{1}{6} a^{3} - \frac{12}{25}$, $\frac{1}{14400} a^{16} + \frac{1}{14400} a^{15} - \frac{1}{1440} a^{14} - \frac{7}{7200} a^{13} + \frac{1}{2400} a^{12} + \frac{7}{360} a^{11} + \frac{7}{900} a^{10} - \frac{7}{225} a^{9} + \frac{1}{40} a^{8} + \frac{11}{90} a^{7} + \frac{13}{180} a^{6} - \frac{13}{180} a^{5} - \frac{1}{10} a^{4} + \frac{2}{45} a^{3} - \frac{22}{45} a^{2} + \frac{83}{450} a + \frac{73}{450}$, $\frac{1}{14400} a^{17} + \frac{1}{14400} a^{15} - \frac{1}{3600} a^{14} - \frac{1}{1440} a^{13} - \frac{7}{7200} a^{12} - \frac{7}{600} a^{11} + \frac{7}{360} a^{10} + \frac{7}{900} a^{9} + \frac{7}{72} a^{8} + \frac{1}{40} a^{7} + \frac{11}{90} a^{6} - \frac{1}{36} a^{5} - \frac{13}{180} a^{4} - \frac{1}{10} a^{3} - \frac{49}{150} a^{2} - \frac{22}{45} a + \frac{83}{450}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $9$
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:  \( 615728.7461734295 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{29}) \), 3.1.87.1, 3.1.108.1, 6.2.219501.1, 6.2.284473296.3 x2, 6.2.284473296.1, 9.1.92169347904.2

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

Sibling fields

Galois closure: data not computed
Degree 6 sibling: data not computed
Degree 9 sibling: data not computed
Degree 12 sibling: 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 R R ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$3.6.6.3$x^{6} + 3 x^{4} + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
3.12.14.6$x^{12} + 3 x^{11} + 3 x^{10} - 6 x^{9} + 3 x^{8} + 9 x^{7} + 9 x^{4} + 9 x^{3} + 9$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$