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Properties

Label 18.2.79428004658...0000.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{12}\cdot 3^{9}\cdot 5^{12}\cdot 7^{9}$
Root discriminant $21.27$
Ramified primes $2, 3, 5, 7$
Class number $1$
Class group Trivial
Galois group $S_3^2$ (as 18T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15, -60, 70, -46, 120, -397, 691, -682, 286, 172, -309, 124, 143, -280, 238, -125, 43, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 43*x^16 - 125*x^15 + 238*x^14 - 280*x^13 + 143*x^12 + 124*x^11 - 309*x^10 + 172*x^9 + 286*x^8 - 682*x^7 + 691*x^6 - 397*x^5 + 120*x^4 - 46*x^3 + 70*x^2 - 60*x + 15)
 
gp: K = bnfinit(x^18 - 9*x^17 + 43*x^16 - 125*x^15 + 238*x^14 - 280*x^13 + 143*x^12 + 124*x^11 - 309*x^10 + 172*x^9 + 286*x^8 - 682*x^7 + 691*x^6 - 397*x^5 + 120*x^4 - 46*x^3 + 70*x^2 - 60*x + 15, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 43 x^{16} - 125 x^{15} + 238 x^{14} - 280 x^{13} + 143 x^{12} + 124 x^{11} - 309 x^{10} + 172 x^{9} + 286 x^{8} - 682 x^{7} + 691 x^{6} - 397 x^{5} + 120 x^{4} - 46 x^{3} + 70 x^{2} - 60 x + 15 \)

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:  \(794280046581000000000000=2^{12}\cdot 3^{9}\cdot 5^{12}\cdot 7^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.27$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{8} + \frac{1}{9} a^{7} + \frac{2}{9} a^{6} - \frac{1}{9} a^{4} + \frac{2}{9} a^{3} - \frac{4}{9} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{13} - \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{2}{9} a^{7} - \frac{1}{9} a^{5} + \frac{2}{9} a^{4} - \frac{4}{9} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{16} - \frac{1}{9} a^{13} - \frac{1}{9} a^{11} + \frac{2}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{3} a^{8} + \frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{2}{9} a^{5} + \frac{4}{9} a^{4} - \frac{4}{9} a^{3} - \frac{1}{9} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{1734092166805209} a^{17} + \frac{3966867473014}{1734092166805209} a^{16} + \frac{18003246921806}{1734092166805209} a^{15} + \frac{25078056793163}{1734092166805209} a^{14} - \frac{35825798214323}{578030722268403} a^{13} - \frac{24304993036782}{192676907422801} a^{12} - \frac{48358750206875}{192676907422801} a^{11} + \frac{56025378673024}{192676907422801} a^{10} + \frac{82570518123813}{192676907422801} a^{9} + \frac{276384959169295}{578030722268403} a^{8} + \frac{61312302310797}{192676907422801} a^{7} + \frac{258720316536617}{578030722268403} a^{6} + \frac{59146294694509}{1734092166805209} a^{5} + \frac{122989479851329}{1734092166805209} a^{4} + \frac{314114477622305}{1734092166805209} a^{3} + \frac{195629833487822}{1734092166805209} a^{2} - \frac{250621617783308}{578030722268403} a - \frac{60193117597727}{578030722268403}$

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:  $9$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 114231.66813442291 \)
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

\(\Q(\sqrt{21}) \), 3.1.300.1, 3.1.175.1, 6.2.3704400.2 x2, 6.2.92610000.1, 6.2.5788125.1, 9.1.9261000000.1

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 R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$5$5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$7$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$