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Properties

Label 18.0.27206534396...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 5^{12}\cdot 67^{9}$
Root discriminant $37.99$
Ramified primes $2, 5, 67$
Class number $12$ (GRH)
Class group $[2, 6]$ (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![100, -100, 1300, -5480, 12020, -17490, 18411, -15497, 11689, -7980, 4705, -1907, 564, -23, -45, 60, -1, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - x^16 + 60*x^15 - 45*x^14 - 23*x^13 + 564*x^12 - 1907*x^11 + 4705*x^10 - 7980*x^9 + 11689*x^8 - 15497*x^7 + 18411*x^6 - 17490*x^5 + 12020*x^4 - 5480*x^3 + 1300*x^2 - 100*x + 100)
 
gp: K = bnfinit(x^18 - 3*x^17 - x^16 + 60*x^15 - 45*x^14 - 23*x^13 + 564*x^12 - 1907*x^11 + 4705*x^10 - 7980*x^9 + 11689*x^8 - 15497*x^7 + 18411*x^6 - 17490*x^5 + 12020*x^4 - 5480*x^3 + 1300*x^2 - 100*x + 100, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - x^{16} + 60 x^{15} - 45 x^{14} - 23 x^{13} + 564 x^{12} - 1907 x^{11} + 4705 x^{10} - 7980 x^{9} + 11689 x^{8} - 15497 x^{7} + 18411 x^{6} - 17490 x^{5} + 12020 x^{4} - 5480 x^{3} + 1300 x^{2} - 100 x + 100 \)

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:  \(-27206534396294947000000000000=-\,2^{12}\cdot 5^{12}\cdot 67^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.99$
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:  $2, 5, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{15} a^{9} + \frac{1}{15} a^{8} - \frac{1}{15} a^{7} + \frac{1}{15} a^{6} + \frac{1}{15} a^{5} + \frac{2}{5} a^{4} + \frac{4}{15} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{30} a^{10} - \frac{1}{30} a^{9} + \frac{1}{15} a^{8} - \frac{1}{15} a^{7} + \frac{2}{15} a^{6} - \frac{1}{5} a^{5} + \frac{7}{30} a^{4} - \frac{13}{30} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{30} a^{11} - \frac{1}{30} a^{9} - \frac{1}{15} a^{8} + \frac{2}{15} a^{7} - \frac{2}{15} a^{6} - \frac{1}{30} a^{5} + \frac{2}{5} a^{4} - \frac{11}{30} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{90} a^{12} + \frac{1}{90} a^{11} - \frac{1}{90} a^{10} - \frac{1}{90} a^{9} + \frac{7}{45} a^{8} - \frac{1}{45} a^{7} - \frac{13}{90} a^{6} + \frac{23}{90} a^{5} - \frac{7}{90} a^{4} + \frac{37}{90} a^{3} - \frac{4}{9} a^{2} - \frac{4}{9} a + \frac{1}{9}$, $\frac{1}{90} a^{13} + \frac{1}{90} a^{11} - \frac{2}{45} a^{8} + \frac{13}{90} a^{7} + \frac{2}{15} a^{6} - \frac{1}{2} a^{5} + \frac{4}{45} a^{4} + \frac{11}{45} a^{3} - \frac{1}{3} a^{2} - \frac{1}{9} a - \frac{4}{9}$, $\frac{1}{540} a^{14} - \frac{1}{180} a^{13} - \frac{7}{540} a^{11} - \frac{1}{108} a^{10} + \frac{1}{45} a^{9} - \frac{19}{540} a^{8} - \frac{61}{540} a^{7} - \frac{37}{270} a^{6} - \frac{1}{4} a^{5} + \frac{83}{540} a^{4} - \frac{29}{270} a^{3} - \frac{2}{9} a^{2} - \frac{1}{6} a + \frac{4}{27}$, $\frac{1}{1620} a^{15} - \frac{1}{1620} a^{14} - \frac{1}{1620} a^{12} - \frac{5}{324} a^{11} - \frac{11}{810} a^{10} - \frac{37}{1620} a^{9} + \frac{83}{540} a^{8} - \frac{47}{810} a^{7} + \frac{179}{1620} a^{6} - \frac{161}{324} a^{5} - \frac{2}{27} a^{4} - \frac{139}{810} a^{3} - \frac{13}{54} a^{2} + \frac{25}{81} a - \frac{28}{81}$, $\frac{1}{108540} a^{16} - \frac{1}{7236} a^{15} - \frac{29}{54270} a^{14} - \frac{379}{108540} a^{13} + \frac{5}{21708} a^{12} + \frac{731}{54270} a^{11} + \frac{1081}{108540} a^{10} - \frac{511}{108540} a^{9} + \frac{730}{5427} a^{8} - \frac{9143}{108540} a^{7} + \frac{2629}{108540} a^{6} + \frac{742}{5427} a^{5} - \frac{8186}{27135} a^{4} - \frac{751}{5427} a^{3} + \frac{1981}{5427} a^{2} + \frac{295}{603} a - \frac{1606}{5427}$, $\frac{1}{66172248386100} a^{17} - \frac{1020629}{327585388050} a^{16} + \frac{925669039}{66172248386100} a^{15} - \frac{1872613453}{2646889935444} a^{14} + \frac{2588994571}{735247204290} a^{13} + \frac{21062839819}{22057416128700} a^{12} - \frac{325848425497}{22057416128700} a^{11} + \frac{27361205399}{33086124193050} a^{10} + \frac{67113682609}{13234449677220} a^{9} - \frac{29962585265}{882296645148} a^{8} - \frac{5913557437}{3676236021450} a^{7} - \frac{5678828081777}{66172248386100} a^{6} + \frac{564179791193}{33086124193050} a^{5} + \frac{299625047750}{661722483861} a^{4} + \frac{381030734911}{3308612419305} a^{3} + \frac{727983472918}{3308612419305} a^{2} - \frac{152314340185}{661722483861} a + \frac{31419220981}{73524720429}$

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

Class group and class number

$C_{2}\times C_{6}$, 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:  \( 2265666.46003 \) (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

\(\Q(\sqrt{-67}) \), 3.1.6700.2 x3, 3.1.6700.1 x3, 3.1.1675.1 x3, 3.1.268.1 x3, 6.0.3007630000.1, 6.0.3007630000.2, 6.0.187976875.1, 6.0.4812208.1, 9.1.20151121000000.1 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 R ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$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$$[\ ]_{3}^{2}$
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$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}$
$67$67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$