↑

Properties

Label 18.18.4469642099...3001.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{3}\cdot 7^{13}\cdot 181^{3}\cdot 257^{6}$
Root discriminant $74.04$
Ramified primes $3, 7, 181, 257$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 18T176

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-22133, 79619, 685681, -3172986, 3012697, 1863610, -3884238, 785608, 1226686, -547108, -135808, 106986, 930, -9422, 821, 387, -53, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 53*x^16 + 387*x^15 + 821*x^14 - 9422*x^13 + 930*x^12 + 106986*x^11 - 135808*x^10 - 547108*x^9 + 1226686*x^8 + 785608*x^7 - 3884238*x^6 + 1863610*x^5 + 3012697*x^4 - 3172986*x^3 + 685681*x^2 + 79619*x - 22133)
 
gp: K = bnfinit(x^18 - 6*x^17 - 53*x^16 + 387*x^15 + 821*x^14 - 9422*x^13 + 930*x^12 + 106986*x^11 - 135808*x^10 - 547108*x^9 + 1226686*x^8 + 785608*x^7 - 3884238*x^6 + 1863610*x^5 + 3012697*x^4 - 3172986*x^3 + 685681*x^2 + 79619*x - 22133, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 53 x^{16} + 387 x^{15} + 821 x^{14} - 9422 x^{13} + 930 x^{12} + 106986 x^{11} - 135808 x^{10} - 547108 x^{9} + 1226686 x^{8} + 785608 x^{7} - 3884238 x^{6} + 1863610 x^{5} + 3012697 x^{4} - 3172986 x^{3} + 685681 x^{2} + 79619 x - 22133 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4469642099029414895974730306243001=3^{3}\cdot 7^{13}\cdot 181^{3}\cdot 257^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.04$
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:  $3, 7, 181, 257$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{14} a^{11} + \frac{1}{7} a^{10} - \frac{1}{7} a^{9} + \frac{1}{7} a^{8} + \frac{1}{7} a^{7} - \frac{1}{7} a^{6} - \frac{5}{14} a^{4} + \frac{1}{7} a^{3} - \frac{3}{7} a^{2} + \frac{2}{7}$, $\frac{1}{14} a^{12} + \frac{1}{14} a^{10} - \frac{1}{14} a^{9} - \frac{1}{7} a^{8} + \frac{1}{14} a^{7} + \frac{2}{7} a^{6} - \frac{5}{14} a^{5} - \frac{1}{7} a^{4} - \frac{3}{14} a^{3} + \frac{5}{14} a^{2} + \frac{2}{7} a - \frac{1}{14}$, $\frac{1}{14} a^{13} - \frac{3}{14} a^{10} - \frac{1}{14} a^{8} + \frac{1}{7} a^{7} - \frac{3}{14} a^{6} - \frac{1}{7} a^{5} + \frac{1}{7} a^{4} + \frac{3}{14} a^{3} - \frac{2}{7} a^{2} - \frac{1}{14} a - \frac{2}{7}$, $\frac{1}{84} a^{14} + \frac{1}{42} a^{13} + \frac{1}{42} a^{12} + \frac{1}{42} a^{10} - \frac{1}{42} a^{9} - \frac{1}{7} a^{8} + \frac{4}{21} a^{7} + \frac{11}{42} a^{6} + \frac{5}{14} a^{5} - \frac{1}{7} a^{4} + \frac{1}{42} a^{3} + \frac{3}{14} a^{2} + \frac{4}{21} a - \frac{5}{84}$, $\frac{1}{168} a^{15} - \frac{1}{168} a^{14} - \frac{1}{42} a^{13} - \frac{1}{42} a^{11} - \frac{1}{12} a^{10} - \frac{1}{12} a^{8} - \frac{4}{21} a^{7} - \frac{1}{2} a^{6} - \frac{2}{7} a^{5} - \frac{1}{6} a^{4} + \frac{11}{28} a^{3} - \frac{1}{3} a^{2} - \frac{71}{168} a - \frac{5}{56}$, $\frac{1}{3528} a^{16} + \frac{1}{882} a^{15} - \frac{17}{3528} a^{14} + \frac{1}{294} a^{13} + \frac{8}{441} a^{12} - \frac{5}{1764} a^{11} - \frac{31}{252} a^{10} + \frac{439}{1764} a^{9} + \frac{13}{588} a^{8} + \frac{23}{294} a^{7} - \frac{251}{882} a^{6} - \frac{149}{441} a^{5} - \frac{1}{1764} a^{4} - \frac{79}{588} a^{3} + \frac{535}{1176} a^{2} + \frac{3}{28} a - \frac{635}{3528}$, $\frac{1}{948299504366598520461461510472} a^{17} + \frac{12453925456323707793758215}{948299504366598520461461510472} a^{16} + \frac{1074328757754450632468193661}{948299504366598520461461510472} a^{15} - \frac{870287073866605800048924901}{316099834788866173487153836824} a^{14} - \frac{371620017497026589552565065}{237074876091649630115365377618} a^{13} + \frac{12219895702842661289060510251}{474149752183299260230730755236} a^{12} - \frac{4065826403288441755065915713}{237074876091649630115365377618} a^{11} - \frac{10728070659841803807017314322}{118537438045824815057682688809} a^{10} + \frac{2563362512747068297256920529}{11289279813888077624541208458} a^{9} + \frac{7528562108128841192832105739}{52683305798144362247858972804} a^{8} - \frac{33189710322726597299013667055}{237074876091649630115365377618} a^{7} - \frac{95441620042066433996230582711}{237074876091649630115365377618} a^{6} + \frac{173333379396328771024812230219}{474149752183299260230730755236} a^{5} + \frac{37601040449257043632627408831}{79024958697216543371788459206} a^{4} - \frac{40591318130211402158634673529}{105366611596288724495717945608} a^{3} + \frac{109773932098907917329950279167}{316099834788866173487153836824} a^{2} + \frac{268131174695159055381302889025}{948299504366598520461461510472} a - \frac{5733439420619754865326546991}{316099834788866173487153836824}$

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:  $17$
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:  \( 116066378910 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T176:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 576
The 40 conjugacy class representatives for t18n176
Character table for t18n176 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.257.1, 9.9.1997043891857.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.12.0.1$x^{12} - x^{4} - x^{3} - x^{2} + x - 1$$1$$12$$0$$C_{12}$$[\ ]^{12}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.4.2.1$x^{4} + 6335 x^{2} + 10614564$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
257Data not computed