↑

Properties

Label 18.6.17171174753...1888.2
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 7^{12}\cdot 13^{13}$
Root discriminant $37.03$
Ramified primes $2, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times A_4^2$ (as 18T109)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5473, 13429, 24115, -68359, 46905, -18087, 13871, -10530, 2511, 3078, -2901, 745, 52, 35, -78, 16, 13, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 13*x^16 + 16*x^15 - 78*x^14 + 35*x^13 + 52*x^12 + 745*x^11 - 2901*x^10 + 3078*x^9 + 2511*x^8 - 10530*x^7 + 13871*x^6 - 18087*x^5 + 46905*x^4 - 68359*x^3 + 24115*x^2 + 13429*x - 5473)
 
gp: K = bnfinit(x^18 - 7*x^17 + 13*x^16 + 16*x^15 - 78*x^14 + 35*x^13 + 52*x^12 + 745*x^11 - 2901*x^10 + 3078*x^9 + 2511*x^8 - 10530*x^7 + 13871*x^6 - 18087*x^5 + 46905*x^4 - 68359*x^3 + 24115*x^2 + 13429*x - 5473, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 13 x^{16} + 16 x^{15} - 78 x^{14} + 35 x^{13} + 52 x^{12} + 745 x^{11} - 2901 x^{10} + 3078 x^{9} + 2511 x^{8} - 10530 x^{7} + 13871 x^{6} - 18087 x^{5} + 46905 x^{4} - 68359 x^{3} + 24115 x^{2} + 13429 x - 5473 \)

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:  \(17171174753799627467382181888=2^{12}\cdot 7^{12}\cdot 13^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.03$
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:  $2, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{13} - \frac{1}{6} a^{12} + \frac{1}{6} a^{10} + \frac{1}{3} a^{9} + \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a + \frac{1}{6}$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{12} - \frac{1}{6} a^{11} + \frac{1}{6} a^{10} - \frac{1}{2} a^{9} - \frac{1}{3} a^{8} - \frac{1}{2} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{6} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6}$, $\frac{1}{204} a^{16} + \frac{1}{68} a^{15} - \frac{4}{51} a^{14} + \frac{19}{204} a^{13} - \frac{8}{51} a^{12} + \frac{11}{102} a^{11} + \frac{13}{51} a^{10} - \frac{101}{204} a^{9} + \frac{31}{68} a^{8} + \frac{13}{68} a^{7} - \frac{1}{102} a^{6} + \frac{23}{68} a^{5} - \frac{77}{204} a^{4} + \frac{14}{51} a^{3} - \frac{20}{51} a^{2} + \frac{25}{204} a - \frac{19}{204}$, $\frac{1}{2158748649352114704629555186346492} a^{17} + \frac{1019657494663759618724299037275}{1079374324676057352314777593173246} a^{16} + \frac{85270960323391420014201978644509}{2158748649352114704629555186346492} a^{15} + \frac{59085511596527717087606504749477}{719582883117371568209851728782164} a^{14} - \frac{71252830309319906171141336382109}{719582883117371568209851728782164} a^{13} - \frac{110378382944590558530130251385769}{1079374324676057352314777593173246} a^{12} + \frac{495357461843635394030148426016033}{1079374324676057352314777593173246} a^{11} + \frac{1074457948405131890607326499113579}{2158748649352114704629555186346492} a^{10} + \frac{493849523314345580142629287079515}{1079374324676057352314777593173246} a^{9} - \frac{41330527497457788428307084389361}{359791441558685784104925864391082} a^{8} - \frac{424098683941981336438104689381149}{2158748649352114704629555186346492} a^{7} + \frac{227535629023843535895118597664921}{719582883117371568209851728782164} a^{6} + \frac{278794353388518058123683385607909}{1079374324676057352314777593173246} a^{5} - \frac{593960502666480131146579991962835}{2158748649352114704629555186346492} a^{4} + \frac{162618486018358164386865676309651}{539687162338028676157388796586623} a^{3} + \frac{11632009382234287312578837796815}{42328404889257151071167748751892} a^{2} - \frac{176248108501480202145355316130247}{539687162338028676157388796586623} a - \frac{422935071380476213524860074985}{5127669000836376970616520632652}$

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:  $11$
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:  \( 18787092.4399 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times A_4^2$ (as 18T109):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 288
The 32 conjugacy class representatives for $C_2\times A_4^2$
Character table for $C_2\times A_4^2$ is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.169.1, 3.3.8281.2, 3.3.8281.1, 9.9.567869252041.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$$[2, 2]^{6}$
2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
7Data not computed
$13$13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.5.5$x^{6} + 104$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$