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Properties

Label 18.0.45639710758...1875.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,5^{12}\cdot 83^{9}$
Root discriminant $26.64$
Ramified primes $5, 83$
Class number $9$
Class group $[3, 3]$
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![1728, -3024, 5868, -8290, 11517, -13785, 14084, -11437, 7485, -4064, 1933, -905, 465, -232, 109, -35, 12, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 12*x^16 - 35*x^15 + 109*x^14 - 232*x^13 + 465*x^12 - 905*x^11 + 1933*x^10 - 4064*x^9 + 7485*x^8 - 11437*x^7 + 14084*x^6 - 13785*x^5 + 11517*x^4 - 8290*x^3 + 5868*x^2 - 3024*x + 1728)
 
gp: K = bnfinit(x^18 - 3*x^17 + 12*x^16 - 35*x^15 + 109*x^14 - 232*x^13 + 465*x^12 - 905*x^11 + 1933*x^10 - 4064*x^9 + 7485*x^8 - 11437*x^7 + 14084*x^6 - 13785*x^5 + 11517*x^4 - 8290*x^3 + 5868*x^2 - 3024*x + 1728, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 12 x^{16} - 35 x^{15} + 109 x^{14} - 232 x^{13} + 465 x^{12} - 905 x^{11} + 1933 x^{10} - 4064 x^{9} + 7485 x^{8} - 11437 x^{7} + 14084 x^{6} - 13785 x^{5} + 11517 x^{4} - 8290 x^{3} + 5868 x^{2} - 3024 x + 1728 \)

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:  \(-45639710758676856201171875=-\,5^{12}\cdot 83^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.64$
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:  $5, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{9} - \frac{1}{16} a^{8} - \frac{1}{8} a^{7} + \frac{1}{16} a^{6} + \frac{1}{8} a^{5} + \frac{1}{16} a^{4} + \frac{1}{16} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{10} + \frac{1}{16} a^{8} - \frac{1}{16} a^{7} - \frac{1}{16} a^{6} + \frac{3}{16} a^{5} - \frac{1}{8} a^{4} + \frac{3}{16} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{11} + \frac{1}{16} a^{7} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} - \frac{1}{8} a^{4} - \frac{3}{16} a^{3} - \frac{1}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{32} a^{12} + \frac{1}{32} a^{8} - \frac{1}{16} a^{7} - \frac{1}{8} a^{6} - \frac{1}{16} a^{5} - \frac{3}{32} a^{4} - \frac{1}{16} a^{3} - \frac{1}{8} a^{2}$, $\frac{1}{96} a^{13} + \frac{1}{96} a^{12} + \frac{1}{48} a^{11} + \frac{1}{48} a^{10} - \frac{1}{32} a^{9} - \frac{1}{32} a^{8} + \frac{1}{48} a^{7} - \frac{5}{32} a^{5} + \frac{5}{32} a^{4} + \frac{1}{16} a^{3} + \frac{5}{24} a^{2} + \frac{1}{6} a$, $\frac{1}{192} a^{14} - \frac{1}{192} a^{13} - \frac{1}{64} a^{12} + \frac{1}{48} a^{11} + \frac{5}{192} a^{10} - \frac{1}{64} a^{9} + \frac{23}{192} a^{8} - \frac{1}{12} a^{7} + \frac{5}{64} a^{6} + \frac{9}{64} a^{5} + \frac{13}{64} a^{4} + \frac{1}{96} a^{3} + \frac{7}{16} a^{2} - \frac{5}{12} a$, $\frac{1}{6720} a^{15} + \frac{1}{2240} a^{14} + \frac{1}{2240} a^{13} - \frac{1}{168} a^{12} + \frac{137}{6720} a^{11} - \frac{11}{6720} a^{10} - \frac{11}{1344} a^{9} + \frac{71}{840} a^{8} - \frac{71}{960} a^{7} + \frac{121}{2240} a^{6} - \frac{493}{2240} a^{5} + \frac{631}{3360} a^{4} - \frac{209}{840} a^{3} + \frac{3}{280} a^{2} + \frac{209}{420} a - \frac{8}{35}$, $\frac{1}{483840} a^{16} + \frac{1}{40320} a^{15} - \frac{61}{32256} a^{14} + \frac{583}{120960} a^{13} + \frac{3581}{241920} a^{12} + \frac{1381}{241920} a^{11} + \frac{7}{7680} a^{10} + \frac{313}{30240} a^{9} - \frac{4883}{48384} a^{8} + \frac{5707}{48384} a^{7} + \frac{1891}{161280} a^{6} + \frac{5921}{30240} a^{5} + \frac{73001}{483840} a^{4} + \frac{20149}{80640} a^{3} - \frac{7729}{40320} a^{2} + \frac{391}{864} a + \frac{523}{2520}$, $\frac{1}{90857894400} a^{17} + \frac{577}{3365107200} a^{16} + \frac{4567}{93187584} a^{15} + \frac{16898771}{18171578880} a^{14} + \frac{179468327}{45428947200} a^{13} - \frac{13184129}{1622462400} a^{12} - \frac{708165011}{30285964800} a^{11} - \frac{216776201}{90857894400} a^{10} - \frac{1093772767}{45428947200} a^{9} + \frac{734760157}{22714473600} a^{8} + \frac{370199833}{3365107200} a^{7} + \frac{184404301}{2595939840} a^{6} + \frac{711655613}{6989068800} a^{5} + \frac{628952839}{3365107200} a^{4} + \frac{255223639}{2163283200} a^{3} - \frac{8839308481}{22714473600} a^{2} - \frac{7420667}{29121120} a + \frac{1151393}{22534200}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 226753.492401 \)
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

\(\Q(\sqrt{-83}) \), 3.1.2075.2 x3, 3.1.2075.1 x3, 3.1.2075.3 x3, 3.1.83.1 x3, 6.0.357366875.3, 6.0.357366875.2, 6.0.357366875.1, 6.0.571787.1, 9.1.741536265625.1 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$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$$[\ ]_{3}^{2}$
$83$83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$