↑

Properties

Label 18.6.12860748263...3856.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 7^{12}\cdot 197^{8}$
Root discriminant $60.79$
Ramified primes $2, 7, 197$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group $C_3:S_4$ (as 18T37)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-60831, 236088, 548514, 8295, -277423, -9361, 12556, 22738, 24077, -12539, -3494, 1276, 219, 332, -110, -42, 0, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 42*x^15 - 110*x^14 + 332*x^13 + 219*x^12 + 1276*x^11 - 3494*x^10 - 12539*x^9 + 24077*x^8 + 22738*x^7 + 12556*x^6 - 9361*x^5 - 277423*x^4 + 8295*x^3 + 548514*x^2 + 236088*x - 60831)
 
gp: K = bnfinit(x^18 - 3*x^17 - 42*x^15 - 110*x^14 + 332*x^13 + 219*x^12 + 1276*x^11 - 3494*x^10 - 12539*x^9 + 24077*x^8 + 22738*x^7 + 12556*x^6 - 9361*x^5 - 277423*x^4 + 8295*x^3 + 548514*x^2 + 236088*x - 60831, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 42 x^{15} - 110 x^{14} + 332 x^{13} + 219 x^{12} + 1276 x^{11} - 3494 x^{10} - 12539 x^{9} + 24077 x^{8} + 22738 x^{7} + 12556 x^{6} - 9361 x^{5} - 277423 x^{4} + 8295 x^{3} + 548514 x^{2} + 236088 x - 60831 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(128607482636630365539205392633856=2^{12}\cdot 7^{12}\cdot 197^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.79$
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:  $2, 7, 197$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{15} a^{11} + \frac{1}{15} a^{10} - \frac{2}{15} a^{9} - \frac{2}{15} a^{8} + \frac{2}{15} a^{7} + \frac{7}{15} a^{6} + \frac{2}{15} a^{5} + \frac{7}{15} a^{4} + \frac{1}{15} a^{3} + \frac{2}{5} a^{2} + \frac{4}{15} a - \frac{2}{5}$, $\frac{1}{15} a^{12} + \frac{2}{15} a^{10} - \frac{1}{15} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{4}{15} a^{4} - \frac{1}{3} a^{3} + \frac{1}{5} a^{2} - \frac{1}{3} a + \frac{2}{5}$, $\frac{1}{15} a^{13} - \frac{2}{15} a^{10} - \frac{2}{15} a^{9} - \frac{1}{15} a^{8} + \frac{1}{15} a^{7} - \frac{4}{15} a^{6} - \frac{4}{15} a^{4} + \frac{1}{15} a^{3} - \frac{2}{15} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{45} a^{14} - \frac{1}{45} a^{13} + \frac{1}{45} a^{11} + \frac{1}{15} a^{10} - \frac{1}{9} a^{9} - \frac{4}{45} a^{8} - \frac{14}{45} a^{7} + \frac{2}{9} a^{6} - \frac{13}{45} a^{5} - \frac{19}{45} a^{4} + \frac{8}{45} a^{2} + \frac{2}{15} a$, $\frac{1}{45} a^{15} - \frac{1}{45} a^{13} + \frac{1}{45} a^{12} + \frac{1}{45} a^{11} - \frac{1}{9} a^{10} - \frac{1}{15} a^{9} + \frac{1}{15} a^{8} + \frac{4}{9} a^{7} - \frac{1}{5} a^{6} - \frac{8}{45} a^{5} + \frac{4}{9} a^{4} - \frac{2}{9} a^{3} + \frac{11}{45} a^{2} - \frac{7}{15} a + \frac{2}{5}$, $\frac{1}{135} a^{16} + \frac{1}{135} a^{15} + \frac{1}{135} a^{14} + \frac{4}{135} a^{13} + \frac{2}{135} a^{12} + \frac{4}{135} a^{11} + \frac{22}{135} a^{10} + \frac{11}{135} a^{9} + \frac{4}{45} a^{8} - \frac{59}{135} a^{7} - \frac{2}{5} a^{6} - \frac{62}{135} a^{5} - \frac{8}{27} a^{4} - \frac{2}{135} a^{3} + \frac{4}{9} a^{2} - \frac{1}{15} a + \frac{2}{5}$, $\frac{1}{24817811607037843073222439235324800434625} a^{17} - \frac{374473961809671092581283530963503401}{551506924600840957182720871896106676325} a^{16} + \frac{550880762036680123006164694704039487}{551506924600840957182720871896106676325} a^{15} - \frac{20838439249805178577747053079161437474}{8272603869012614357740813078441600144875} a^{14} - \frac{211519562182005910015531171588619475236}{24817811607037843073222439235324800434625} a^{13} - \frac{784242067461703481432093960762848103506}{24817811607037843073222439235324800434625} a^{12} + \frac{19231602601406427784895894111891105473}{919178207668068261971201453160177793875} a^{11} + \frac{3706291907899329277467245744136564244369}{24817811607037843073222439235324800434625} a^{10} - \frac{2531346639328282105459078247753978303492}{24817811607037843073222439235324800434625} a^{9} - \frac{141228741688393373546817860299803790229}{992712464281513722928897569412992017385} a^{8} - \frac{11415814052893366278215256720161573741323}{24817811607037843073222439235324800434625} a^{7} - \frac{2991934232794563231234769856839759071671}{24817811607037843073222439235324800434625} a^{6} - \frac{7991774021267511685785229873402195583837}{24817811607037843073222439235324800434625} a^{5} - \frac{7723307837623354896905139374376292378132}{24817811607037843073222439235324800434625} a^{4} + \frac{10458353600099435894700755107045742372396}{24817811607037843073222439235324800434625} a^{3} - \frac{472582278546783605487613689463749189329}{8272603869012614357740813078441600144875} a^{2} + \frac{593132930860532797870456077605865188852}{2757534623004204785913604359480533381625} a + \frac{343768254202348486610463512738720836591}{919178207668068261971201453160177793875}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  $11$
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:  \( 1033559423.06 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_4$ (as 18T37):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 9 conjugacy class representatives for $C_3:S_4$
Character table for $C_3:S_4$

Intermediate fields

3.3.9653.1, 3.3.788.1, 3.3.38612.2, 3.3.38612.1, 6.2.93180409.3, 9.9.11340523913674816.1

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

Sibling fields

Degree 12 siblings: 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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
197Data not computed