↑

Properties

Label 18.6.11247392561...9689.2
Degree $18$
Signature $[6, 6]$
Discriminant $3^{24}\cdot 73^{5}\cdot 577^{3}$
Root discriminant $41.11$
Ramified primes $3, 73, 577$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group 18T401

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![289, 3366, 2862, -24970, 84738, -148209, 161951, -119193, 60993, -18927, -1518, 5931, -3982, 1455, -258, -34, 33, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 33*x^16 - 34*x^15 - 258*x^14 + 1455*x^13 - 3982*x^12 + 5931*x^11 - 1518*x^10 - 18927*x^9 + 60993*x^8 - 119193*x^7 + 161951*x^6 - 148209*x^5 + 84738*x^4 - 24970*x^3 + 2862*x^2 + 3366*x + 289)
 
gp: K = bnfinit(x^18 - 9*x^17 + 33*x^16 - 34*x^15 - 258*x^14 + 1455*x^13 - 3982*x^12 + 5931*x^11 - 1518*x^10 - 18927*x^9 + 60993*x^8 - 119193*x^7 + 161951*x^6 - 148209*x^5 + 84738*x^4 - 24970*x^3 + 2862*x^2 + 3366*x + 289, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 33 x^{16} - 34 x^{15} - 258 x^{14} + 1455 x^{13} - 3982 x^{12} + 5931 x^{11} - 1518 x^{10} - 18927 x^{9} + 60993 x^{8} - 119193 x^{7} + 161951 x^{6} - 148209 x^{5} + 84738 x^{4} - 24970 x^{3} + 2862 x^{2} + 3366 x + 289 \)

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:  \(112473925614060022797462679689=3^{24}\cdot 73^{5}\cdot 577^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.11$
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:  $3, 73, 577$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{73} a^{16} + \frac{35}{73} a^{15} - \frac{6}{73} a^{14} - \frac{10}{73} a^{13} + \frac{16}{73} a^{12} - \frac{9}{73} a^{11} - \frac{4}{73} a^{10} - \frac{36}{73} a^{9} + \frac{2}{73} a^{8} - \frac{28}{73} a^{7} + \frac{28}{73} a^{6} - \frac{19}{73} a^{5} + \frac{30}{73} a^{4} - \frac{15}{73} a^{3} - \frac{11}{73} a^{2} - \frac{17}{73} a - \frac{8}{73}$, $\frac{1}{426580905349178275192413142266669941261} a^{17} - \frac{2583246164054809440837878599058140645}{426580905349178275192413142266669941261} a^{16} + \frac{1239453488355589424277222628870847250}{426580905349178275192413142266669941261} a^{15} + \frac{2892583380560921677519347202792751598}{25092994432304604423083126015686467133} a^{14} + \frac{202266013055783063745617908213598706644}{426580905349178275192413142266669941261} a^{13} - \frac{136766461934331688443578700332621542386}{426580905349178275192413142266669941261} a^{12} - \frac{4515770522859971521521387243188316127}{426580905349178275192413142266669941261} a^{11} - \frac{128433174055194674789165367444547051859}{426580905349178275192413142266669941261} a^{10} + \frac{113102548549634603862078427601621737484}{426580905349178275192413142266669941261} a^{9} - \frac{56001531550141656374280894589489712397}{426580905349178275192413142266669941261} a^{8} + \frac{35239466384951152532975690302330499050}{426580905349178275192413142266669941261} a^{7} + \frac{15126437643916999502002017974140279440}{426580905349178275192413142266669941261} a^{6} - \frac{147225615291529834454903288875800969246}{426580905349178275192413142266669941261} a^{5} + \frac{191926744639923047722752050763989870640}{426580905349178275192413142266669941261} a^{4} + \frac{165244097413651033088583823386172089995}{426580905349178275192413142266669941261} a^{3} - \frac{28542877952490454695374591095119938791}{426580905349178275192413142266669941261} a^{2} + \frac{196058216639101667000002122746306866700}{426580905349178275192413142266669941261} a + \frac{10701530948572392675258667990496220802}{25092994432304604423083126015686467133}$

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:  $11$
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:  \( 10371943.796 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T401:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2592
The 44 conjugacy class representatives for t18n401
Character table for t18n401 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 6.2.276355881.1, 9.9.22384826361.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
73Data not computed
577Data not computed