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Properties

Label 18.0.39522986474...3927.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 7^{12}\cdot 29^{9}$
Root discriminant $34.13$
Ramified primes $3, 7, 29$
Class number $6$ (GRH)
Class group $[6]$ (GRH)
Galois group $D_9$ (as 18T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![391959, 50868, -131238, 148365, -33918, -73656, 64117, -27555, -3273, 9969, -4680, 693, 736, -651, 321, -132, 42, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 42*x^16 - 132*x^15 + 321*x^14 - 651*x^13 + 736*x^12 + 693*x^11 - 4680*x^10 + 9969*x^9 - 3273*x^8 - 27555*x^7 + 64117*x^6 - 73656*x^5 - 33918*x^4 + 148365*x^3 - 131238*x^2 + 50868*x + 391959)
 
gp: K = bnfinit(x^18 - 9*x^17 + 42*x^16 - 132*x^15 + 321*x^14 - 651*x^13 + 736*x^12 + 693*x^11 - 4680*x^10 + 9969*x^9 - 3273*x^8 - 27555*x^7 + 64117*x^6 - 73656*x^5 - 33918*x^4 + 148365*x^3 - 131238*x^2 + 50868*x + 391959, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 42 x^{16} - 132 x^{15} + 321 x^{14} - 651 x^{13} + 736 x^{12} + 693 x^{11} - 4680 x^{10} + 9969 x^{9} - 3273 x^{8} - 27555 x^{7} + 64117 x^{6} - 73656 x^{5} - 33918 x^{4} + 148365 x^{3} - 131238 x^{2} + 50868 x + 391959 \)

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:  \(-3952298647444414435530183927=-\,3^{9}\cdot 7^{12}\cdot 29^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.13$
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:  $3, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{5} - \frac{1}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{10} + \frac{2}{9} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{45} a^{11} + \frac{2}{45} a^{10} - \frac{2}{45} a^{9} - \frac{2}{15} a^{8} + \frac{2}{15} a^{7} + \frac{2}{15} a^{6} - \frac{4}{45} a^{5} + \frac{1}{45} a^{4} + \frac{1}{9} a^{3} - \frac{7}{15} a^{2} + \frac{1}{15} a - \frac{2}{5}$, $\frac{1}{45} a^{12} - \frac{1}{45} a^{10} - \frac{2}{45} a^{9} + \frac{1}{15} a^{8} - \frac{2}{15} a^{7} - \frac{1}{45} a^{6} + \frac{1}{5} a^{5} - \frac{17}{45} a^{4} + \frac{14}{45} a^{3} + \frac{1}{3} a^{2} + \frac{7}{15} a - \frac{1}{5}$, $\frac{1}{45} a^{13} + \frac{1}{45} a^{9} + \frac{1}{15} a^{8} + \frac{1}{9} a^{7} - \frac{7}{15} a^{5} + \frac{4}{9} a^{3} + \frac{1}{3} a^{2} - \frac{2}{15} a - \frac{2}{5}$, $\frac{1}{135} a^{14} - \frac{1}{135} a^{13} - \frac{1}{135} a^{12} + \frac{1}{135} a^{11} - \frac{2}{45} a^{10} - \frac{1}{45} a^{9} - \frac{7}{135} a^{8} + \frac{7}{135} a^{7} - \frac{14}{135} a^{6} - \frac{67}{135} a^{5} + \frac{7}{15} a^{4} - \frac{2}{5} a^{3} + \frac{2}{15} a^{2} - \frac{1}{3} a + \frac{2}{5}$, $\frac{1}{135} a^{15} + \frac{1}{135} a^{13} + \frac{1}{135} a^{11} + \frac{1}{45} a^{10} - \frac{4}{135} a^{9} + \frac{2}{15} a^{8} - \frac{1}{135} a^{7} - \frac{46}{135} a^{5} + \frac{4}{9} a^{4} - \frac{2}{45} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{9923641941752325} a^{16} - \frac{8}{9923641941752325} a^{15} - \frac{360405774669}{122514098046325} a^{14} - \frac{16175302245922}{9923641941752325} a^{13} - \frac{36382407332029}{9923641941752325} a^{12} - \frac{154041725654}{122514098046325} a^{11} + \frac{220453484150144}{9923641941752325} a^{10} + \frac{96766366909754}{9923641941752325} a^{9} - \frac{1681753406686}{3307880647250775} a^{8} - \frac{1107914960723636}{9923641941752325} a^{7} + \frac{195341221014929}{1984728388350465} a^{6} + \frac{1000862421457696}{3307880647250775} a^{5} + \frac{154609033904312}{661576129450155} a^{4} - \frac{48285463944887}{220525376483385} a^{3} + \frac{7673097127}{13957302309075} a^{2} + \frac{26977278931076}{367542294138975} a + \frac{56287281189473}{122514098046325}$, $\frac{1}{22953383811273127725} a^{17} + \frac{1148}{22953383811273127725} a^{16} + \frac{62158963300557133}{22953383811273127725} a^{15} + \frac{10121419501041209}{22953383811273127725} a^{14} + \frac{61536205236334279}{22953383811273127725} a^{13} - \frac{251569647914819248}{22953383811273127725} a^{12} + \frac{298458994871797}{58109832433602855} a^{11} - \frac{1099305342939991067}{22953383811273127725} a^{10} + \frac{462933274249004486}{22953383811273127725} a^{9} - \frac{1848327129411375434}{22953383811273127725} a^{8} + \frac{173489242534058579}{22953383811273127725} a^{7} - \frac{2481674499702567287}{22953383811273127725} a^{6} + \frac{2753124086314253021}{7651127937091042575} a^{5} + \frac{508665570321880679}{1530225587418208515} a^{4} + \frac{816137655468153143}{2550375979030347525} a^{3} - \frac{210114098618124479}{2550375979030347525} a^{2} - \frac{844650686692226}{11335004351245989} a - \frac{37174574622980689}{94458369593716575}$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( -1 \) (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:  \( 11080835.8701 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_9$ (as 18T5):

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 $D_9$
Character table for $D_9$

Intermediate fields

\(\Q(\sqrt{-87}) \), 3.1.87.1 x3, 6.0.658503.1, 9.1.6740083091889.1 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$7$7.9.6.3$x^{9} - 14 x^{6} + 49 x^{3} - 1372$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
7.9.6.3$x^{9} - 14 x^{6} + 49 x^{3} - 1372$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$