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Properties

Label 18.12.2466930139...1211.2
Degree $18$
Signature $[12, 3]$
Discriminant $-\,7^{17}\cdot 13^{9}$
Root discriminant $22.65$
Ramified primes $7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_3^2.A_4$ (as 18T92)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 11, -8, -143, 29, 651, -70, -1279, 92, 1104, -20, -369, -56, 21, 36, 11, -8, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 8*x^16 + 11*x^15 + 36*x^14 + 21*x^13 - 56*x^12 - 369*x^11 - 20*x^10 + 1104*x^9 + 92*x^8 - 1279*x^7 - 70*x^6 + 651*x^5 + 29*x^4 - 143*x^3 - 8*x^2 + 11*x + 1)
 
gp: K = bnfinit(x^18 - 3*x^17 - 8*x^16 + 11*x^15 + 36*x^14 + 21*x^13 - 56*x^12 - 369*x^11 - 20*x^10 + 1104*x^9 + 92*x^8 - 1279*x^7 - 70*x^6 + 651*x^5 + 29*x^4 - 143*x^3 - 8*x^2 + 11*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 8 x^{16} + 11 x^{15} + 36 x^{14} + 21 x^{13} - 56 x^{12} - 369 x^{11} - 20 x^{10} + 1104 x^{9} + 92 x^{8} - 1279 x^{7} - 70 x^{6} + 651 x^{5} + 29 x^{4} - 143 x^{3} - 8 x^{2} + 11 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2466930139718004361521211=-\,7^{17}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.65$
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:  $7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{1137561655672852061} a^{17} - \frac{490594432031456443}{1137561655672852061} a^{16} + \frac{443112188960428971}{1137561655672852061} a^{15} + \frac{311651291523583189}{1137561655672852061} a^{14} + \frac{461644663589912525}{1137561655672852061} a^{13} - \frac{241478136055602331}{1137561655672852061} a^{12} + \frac{254324055704643524}{1137561655672852061} a^{11} - \frac{499917196576810179}{1137561655672852061} a^{10} - \frac{288272347659932474}{1137561655672852061} a^{9} - \frac{316017032813195194}{1137561655672852061} a^{8} - \frac{267790418597687848}{1137561655672852061} a^{7} - \frac{561466016097148071}{1137561655672852061} a^{6} + \frac{98821680437238466}{1137561655672852061} a^{5} - \frac{161076337043078187}{1137561655672852061} a^{4} - \frac{279588615440725345}{1137561655672852061} a^{3} - \frac{254352868315773419}{1137561655672852061} a^{2} + \frac{203083655306223794}{1137561655672852061} a - \frac{296672975246041944}{1137561655672852061}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $14$
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:  \( 599704.249482 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3^2.A_4$ (as 18T92):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 216
The 40 conjugacy class representatives for $C_2\times C_3^2.A_4$
Character table for $C_2\times C_3^2.A_4$ is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.4.218491.1, 9.9.164648481361.1

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 $18$ $18$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R $18$ R $18$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
7Data not computed
$13$13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.6.5.1$x^{6} - 52$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.4.1$x^{6} + 39 x^{3} + 676$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$