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Properties

Label 18.0.26756573428...7616.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{33}\cdot 3^{8}\cdot 7^{15}$
Root discriminant $29.39$
Ramified primes $2, 3, 7$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group $C_2\times S_3^2$ (as 18T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![526, -2592, 9112, -13880, 11786, -5348, 2237, -3574, 5325, -4778, 2502, -442, -397, 370, -136, 10, 13, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 13*x^16 + 10*x^15 - 136*x^14 + 370*x^13 - 397*x^12 - 442*x^11 + 2502*x^10 - 4778*x^9 + 5325*x^8 - 3574*x^7 + 2237*x^6 - 5348*x^5 + 11786*x^4 - 13880*x^3 + 9112*x^2 - 2592*x + 526)
 
gp: K = bnfinit(x^18 - 6*x^17 + 13*x^16 + 10*x^15 - 136*x^14 + 370*x^13 - 397*x^12 - 442*x^11 + 2502*x^10 - 4778*x^9 + 5325*x^8 - 3574*x^7 + 2237*x^6 - 5348*x^5 + 11786*x^4 - 13880*x^3 + 9112*x^2 - 2592*x + 526, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 13 x^{16} + 10 x^{15} - 136 x^{14} + 370 x^{13} - 397 x^{12} - 442 x^{11} + 2502 x^{10} - 4778 x^{9} + 5325 x^{8} - 3574 x^{7} + 2237 x^{6} - 5348 x^{5} + 11786 x^{4} - 13880 x^{3} + 9112 x^{2} - 2592 x + 526 \)

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:  \(-267565734285752664500207616=-\,2^{33}\cdot 3^{8}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.39$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{5} + \frac{1}{3} a$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{6} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{4}{9} a^{7} + \frac{2}{9} a^{6} - \frac{4}{9} a^{5} - \frac{2}{9} a^{4} - \frac{2}{9} a^{3} - \frac{4}{9} a^{2} + \frac{2}{9} a + \frac{4}{9}$, $\frac{1}{45} a^{12} + \frac{1}{45} a^{11} - \frac{1}{15} a^{10} - \frac{1}{45} a^{9} - \frac{1}{15} a^{8} - \frac{8}{45} a^{7} - \frac{1}{5} a^{6} - \frac{2}{9} a^{5} + \frac{4}{15} a^{4} + \frac{1}{45} a^{3} + \frac{1}{15} a^{2} + \frac{8}{45} a - \frac{1}{45}$, $\frac{1}{135} a^{13} - \frac{1}{135} a^{12} - \frac{1}{27} a^{11} + \frac{1}{27} a^{10} - \frac{1}{135} a^{9} - \frac{2}{135} a^{8} - \frac{38}{135} a^{7} + \frac{53}{135} a^{6} - \frac{58}{135} a^{5} + \frac{67}{135} a^{4} + \frac{46}{135} a^{3} + \frac{47}{135} a^{2} + \frac{28}{135} a + \frac{47}{135}$, $\frac{1}{135} a^{14} + \frac{2}{45} a^{11} - \frac{14}{135} a^{10} - \frac{1}{15} a^{9} - \frac{13}{135} a^{8} - \frac{11}{45} a^{7} - \frac{59}{135} a^{6} - \frac{17}{45} a^{5} - \frac{8}{27} a^{4} - \frac{4}{15} a^{3} - \frac{14}{45} a^{2} - \frac{4}{45} a - \frac{49}{135}$, $\frac{1}{317925} a^{15} + \frac{52}{21195} a^{14} - \frac{1148}{317925} a^{13} + \frac{688}{63585} a^{12} - \frac{9088}{317925} a^{11} - \frac{38512}{317925} a^{10} - \frac{21851}{317925} a^{9} + \frac{3839}{63585} a^{8} + \frac{6782}{317925} a^{7} + \frac{84191}{317925} a^{6} + \frac{117124}{317925} a^{5} + \frac{796}{12717} a^{4} - \frac{69914}{317925} a^{3} - \frac{19673}{63585} a^{2} - \frac{3074}{7065} a - \frac{128302}{317925}$, $\frac{1}{317925} a^{16} + \frac{397}{317925} a^{14} - \frac{146}{63585} a^{13} + \frac{1832}{317925} a^{12} + \frac{11033}{317925} a^{11} - \frac{25226}{317925} a^{10} + \frac{2084}{63585} a^{9} - \frac{20488}{317925} a^{8} - \frac{29479}{317925} a^{7} - \frac{12206}{317925} a^{6} + \frac{20153}{63585} a^{5} - \frac{25364}{317925} a^{4} - \frac{30398}{63585} a^{3} + \frac{746}{21195} a^{2} - \frac{2767}{317925} a + \frac{92}{4239}$, $\frac{1}{303444470025} a^{17} + \frac{26171}{60688894005} a^{16} - \frac{5021}{60688894005} a^{15} - \frac{8209144}{6743210445} a^{14} - \frac{832464112}{303444470025} a^{13} + \frac{648613306}{101148156675} a^{12} - \frac{1475442329}{60688894005} a^{11} - \frac{25819618256}{303444470025} a^{10} + \frac{171720059}{4273865775} a^{9} - \frac{33522780874}{303444470025} a^{8} - \frac{24510797633}{60688894005} a^{7} - \frac{64427773052}{303444470025} a^{6} + \frac{7176909386}{101148156675} a^{5} + \frac{29498966153}{60688894005} a^{4} - \frac{1808490197}{3746228025} a^{3} - \frac{66243597872}{303444470025} a^{2} - \frac{2935793068}{12137778801} a - \frac{72442110016}{303444470025}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  \( 1040005.4350050329 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_3^2$ (as 18T29):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 18 conjugacy class representatives for $C_2\times S_3^2$
Character table for $C_2\times S_3^2$

Intermediate fields

\(\Q(\sqrt{-14}) \), 3.1.1176.1, 3.1.588.1, 6.0.309786624.1, 6.0.77446656.1, 9.1.78066229248.1

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

Sibling fields

Degree 12 siblings: data not computed
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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.6.11.1$x^{6} + 14$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
2.12.22.60$x^{12} - 84 x^{10} + 444 x^{8} + 32 x^{6} - 272 x^{4} - 320 x^{2} + 64$$6$$2$$22$$D_6$$[3]_{3}^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.2$x^{2} + 3$$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}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7Data not computed