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Properties

Label 18.2.77171478574...0944.2
Degree $18$
Signature $[2, 8]$
Discriminant $2^{12}\cdot 3^{21}\cdot 23^{9}$
Root discriminant $27.43$
Ramified primes $2, 3, 23$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
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![997, 3300, 4788, 2971, -798, -3132, -2804, -366, 1848, 1373, -354, -732, -86, 192, 60, -21, -12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 12*x^16 - 21*x^15 + 60*x^14 + 192*x^13 - 86*x^12 - 732*x^11 - 354*x^10 + 1373*x^9 + 1848*x^8 - 366*x^7 - 2804*x^6 - 3132*x^5 - 798*x^4 + 2971*x^3 + 4788*x^2 + 3300*x + 997)
 
gp: K = bnfinit(x^18 - 12*x^16 - 21*x^15 + 60*x^14 + 192*x^13 - 86*x^12 - 732*x^11 - 354*x^10 + 1373*x^9 + 1848*x^8 - 366*x^7 - 2804*x^6 - 3132*x^5 - 798*x^4 + 2971*x^3 + 4788*x^2 + 3300*x + 997, 1)
 

Normalized defining polynomial

\( x^{18} - 12 x^{16} - 21 x^{15} + 60 x^{14} + 192 x^{13} - 86 x^{12} - 732 x^{11} - 354 x^{10} + 1373 x^{9} + 1848 x^{8} - 366 x^{7} - 2804 x^{6} - 3132 x^{5} - 798 x^{4} + 2971 x^{3} + 4788 x^{2} + 3300 x + 997 \)

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:  \(77171478574802807668690944=2^{12}\cdot 3^{21}\cdot 23^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.43$
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:  $2, 3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5} a^{11} - \frac{2}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{10} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{5} a^{15} - \frac{2}{5} a^{10} - \frac{2}{5} a^{9} + \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{25} a^{16} - \frac{2}{25} a^{15} - \frac{2}{25} a^{14} + \frac{1}{25} a^{13} + \frac{1}{25} a^{12} + \frac{1}{25} a^{11} - \frac{7}{25} a^{10} - \frac{7}{25} a^{9} - \frac{7}{25} a^{8} + \frac{1}{5} a^{7} + \frac{1}{25} a^{6} + \frac{7}{25} a^{5} - \frac{7}{25} a^{4} + \frac{4}{25} a^{3} + \frac{12}{25} a^{2} + \frac{11}{25} a - \frac{12}{25}$, $\frac{1}{71266382139595083469325} a^{17} - \frac{1122260733518086069099}{71266382139595083469325} a^{16} + \frac{5504865653766128342167}{71266382139595083469325} a^{15} + \frac{1264120598847581540787}{14253276427919016693865} a^{14} - \frac{644590366662547178956}{71266382139595083469325} a^{13} - \frac{3947301551961274781816}{71266382139595083469325} a^{12} - \frac{6596361354533063815949}{71266382139595083469325} a^{11} - \frac{18626483265474343277318}{71266382139595083469325} a^{10} - \frac{7273721277495792469568}{71266382139595083469325} a^{9} + \frac{11823706081345424963234}{71266382139595083469325} a^{8} - \frac{28860219207764187414859}{71266382139595083469325} a^{7} + \frac{770238181710095436757}{14253276427919016693865} a^{6} + \frac{13948644720824473771604}{71266382139595083469325} a^{5} - \frac{17023128281701963444362}{71266382139595083469325} a^{4} + \frac{20599556505573807881554}{71266382139595083469325} a^{3} + \frac{1812912228885783358781}{4192140125858534321725} a^{2} - \frac{15806257126157982147604}{71266382139595083469325} a + \frac{32455368094514956639474}{71266382139595083469325}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 271982.19349677156 \) (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

\(\Q(\sqrt{69}) \), 3.1.108.1, 3.1.23.1, 6.2.425747664.3, 6.2.425747664.2 x2, 6.2.328509.1, 9.1.15326915904.1

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
2Data not computed
3Data not computed
$23$23.2.1.1$x^{2} - 23$$2$$1$$1$$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}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$