↑

Properties

Label 18.0.45028390589...0000.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{37}\cdot 5^{12}$
Root discriminant $44.40$
Ramified primes $2, 3, 5$
Class number $108$ (GRH)
Class group $[3, 6, 6]$ (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1156, 11016, 121500, -188676, 99414, 11718, -36621, 17991, -423, -5414, 3807, -729, -216, -27, 189, -126, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 126*x^15 + 189*x^14 - 27*x^13 - 216*x^12 - 729*x^11 + 3807*x^10 - 5414*x^9 - 423*x^8 + 17991*x^7 - 36621*x^6 + 11718*x^5 + 99414*x^4 - 188676*x^3 + 121500*x^2 + 11016*x + 1156)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 - 126*x^15 + 189*x^14 - 27*x^13 - 216*x^12 - 729*x^11 + 3807*x^10 - 5414*x^9 - 423*x^8 + 17991*x^7 - 36621*x^6 + 11718*x^5 + 99414*x^4 - 188676*x^3 + 121500*x^2 + 11016*x + 1156, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} - 126 x^{15} + 189 x^{14} - 27 x^{13} - 216 x^{12} - 729 x^{11} + 3807 x^{10} - 5414 x^{9} - 423 x^{8} + 17991 x^{7} - 36621 x^{6} + 11718 x^{5} + 99414 x^{4} - 188676 x^{3} + 121500 x^{2} + 11016 x + 1156 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-450283905890997363000000000000=-\,2^{12}\cdot 3^{37}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.40$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{7} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{45} a^{9} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{7}{45}$, $\frac{1}{45} a^{10} + \frac{1}{5} a^{5} + \frac{4}{9} a - \frac{1}{5}$, $\frac{1}{45} a^{11} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{11}{45} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{450} a^{12} + \frac{2}{225} a^{11} - \frac{2}{225} a^{10} - \frac{1}{90} a^{9} - \frac{2}{25} a^{7} - \frac{1}{50} a^{6} - \frac{7}{25} a^{5} - \frac{1}{5} a^{4} - \frac{1}{18} a^{3} - \frac{77}{225} a^{2} - \frac{13}{225} a - \frac{77}{225}$, $\frac{1}{1350} a^{13} - \frac{1}{1350} a^{12} + \frac{1}{225} a^{11} + \frac{1}{270} a^{10} - \frac{1}{270} a^{9} + \frac{1}{25} a^{8} - \frac{1}{150} a^{7} + \frac{11}{150} a^{6} + \frac{1}{5} a^{5} - \frac{77}{270} a^{4} - \frac{659}{1350} a^{3} + \frac{74}{225} a^{2} + \frac{203}{675} a - \frac{64}{135}$, $\frac{1}{1350} a^{14} - \frac{1}{1350} a^{12} - \frac{13}{1350} a^{11} - \frac{1}{225} a^{10} - \frac{11}{1350} a^{9} + \frac{1}{30} a^{8} + \frac{2}{75} a^{7} - \frac{13}{150} a^{6} + \frac{641}{1350} a^{5} + \frac{17}{75} a^{4} + \frac{41}{270} a^{3} + \frac{77}{675} a^{2} - \frac{68}{225} a - \frac{83}{675}$, $\frac{1}{1350} a^{15} + \frac{1}{1350} a^{12} - \frac{1}{225} a^{10} - \frac{1}{270} a^{9} + \frac{1}{15} a^{8} - \frac{7}{75} a^{7} + \frac{13}{270} a^{6} + \frac{32}{75} a^{5} - \frac{2}{15} a^{4} - \frac{61}{135} a^{3} + \frac{2}{75} a^{2} + \frac{8}{45} a + \frac{62}{135}$, $\frac{1}{83700} a^{16} + \frac{1}{20925} a^{15} + \frac{23}{83700} a^{14} + \frac{1}{83700} a^{13} + \frac{1}{1674} a^{12} - \frac{629}{83700} a^{11} - \frac{623}{83700} a^{10} - \frac{8}{2325} a^{9} - \frac{203}{3100} a^{8} - \frac{1337}{16740} a^{7} + \frac{191}{20925} a^{6} + \frac{34363}{83700} a^{5} - \frac{6268}{20925} a^{4} + \frac{6233}{41850} a^{3} - \frac{763}{8370} a^{2} + \frac{4328}{20925} a + \frac{11}{2325}$, $\frac{1}{21822379695551790900} a^{17} - \frac{59536140920819}{10911189847775895450} a^{16} - \frac{356602764606103}{2424708855061310100} a^{15} - \frac{21177735745259}{2424708855061310100} a^{14} + \frac{2761515339976211}{10911189847775895450} a^{13} + \frac{21719463545146499}{21822379695551790900} a^{12} + \frac{24266015734430069}{21822379695551790900} a^{11} - \frac{15448502593837063}{1818531641295982575} a^{10} + \frac{57674122184656891}{21822379695551790900} a^{9} + \frac{843473418846015143}{21822379695551790900} a^{8} + \frac{441050773781179864}{5455594923887947725} a^{7} + \frac{77852395596481533}{808236285020436700} a^{6} - \frac{13302538200548913}{202059071255109175} a^{5} + \frac{2030771460769268936}{5455594923887947725} a^{4} - \frac{1393983382973593982}{5455594923887947725} a^{3} - \frac{637242093008258}{7039477321145739} a^{2} + \frac{151186901618937373}{1818531641295982575} a - \frac{66126465329720531}{320917348463996925}$

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

Class group and class number

$C_{3}\times C_{6}\times C_{6}$, which has order $108$ (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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{16580722975945}{218223796955517909} a^{17} + \frac{616288653199925}{872895187822071636} a^{16} - \frac{1549449380180957}{436447593911035818} a^{15} + \frac{8822454428403755}{872895187822071636} a^{14} - \frac{13264872400894195}{872895187822071636} a^{13} + \frac{543259332915545}{218223796955517909} a^{12} + \frac{16120294174521395}{872895187822071636} a^{11} + \frac{49458692589293621}{872895187822071636} a^{10} - \frac{132822660357257615}{436447593911035818} a^{9} + \frac{379599875740379095}{872895187822071636} a^{8} + \frac{24401006559029995}{872895187822071636} a^{7} - \frac{617704859030346235}{436447593911035818} a^{6} + \frac{2629026047845200643}{872895187822071636} a^{5} - \frac{218289606409466330}{218223796955517909} a^{4} - \frac{3459436774271394085}{436447593911035818} a^{3} + \frac{6591770227628359775}{436447593911035818} a^{2} - \frac{2089576757630355020}{218223796955517909} a + \frac{54616452757346}{414086901243867} \) (order $6$)
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:  \( 2396291.4248 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3$ (as 18T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.24300.1 x3, 3.1.24300.2 x3, 3.1.108.1 x3, 3.1.6075.2 x3, 6.0.1771470000.3, 6.0.1771470000.4, 6.0.34992.1, 6.0.110716875.2, 9.1.387420489000000.21 x9

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

Sibling fields

Degree 9 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 R R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$2$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
3Data not computed
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$