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Properties

Label 18.12.5851689066...8403.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,7^{12}\cdot 83^{5}\cdot 181^{4}$
Root discriminant $39.64$
Ramified primes $7, 83, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T400

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 16, -266, 305, 1851, -5248, -7126, 14053, 7966, -15266, -1302, 6337, -665, -1025, 103, 116, -14, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 14*x^16 + 116*x^15 + 103*x^14 - 1025*x^13 - 665*x^12 + 6337*x^11 - 1302*x^10 - 15266*x^9 + 7966*x^8 + 14053*x^7 - 7126*x^6 - 5248*x^5 + 1851*x^4 + 305*x^3 - 266*x^2 + 16*x + 1)
 
gp: K = bnfinit(x^18 - 6*x^17 - 14*x^16 + 116*x^15 + 103*x^14 - 1025*x^13 - 665*x^12 + 6337*x^11 - 1302*x^10 - 15266*x^9 + 7966*x^8 + 14053*x^7 - 7126*x^6 - 5248*x^5 + 1851*x^4 + 305*x^3 - 266*x^2 + 16*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 14 x^{16} + 116 x^{15} + 103 x^{14} - 1025 x^{13} - 665 x^{12} + 6337 x^{11} - 1302 x^{10} - 15266 x^{9} + 7966 x^{8} + 14053 x^{7} - 7126 x^{6} - 5248 x^{5} + 1851 x^{4} + 305 x^{3} - 266 x^{2} + 16 x + 1 \)

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:  \(-58516890664476634710877708403=-\,7^{12}\cdot 83^{5}\cdot 181^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.64$
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:  $7, 83, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} + \frac{1}{7} a^{11} - \frac{2}{7} a^{10} + \frac{2}{7} a^{9} + \frac{3}{7} a^{8} + \frac{1}{7} a^{7} + \frac{3}{7} a^{5} + \frac{1}{7} a^{4} - \frac{2}{7} a^{3} + \frac{1}{7} a^{2} - \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{13} - \frac{3}{7} a^{11} - \frac{3}{7} a^{10} + \frac{1}{7} a^{9} - \frac{2}{7} a^{8} - \frac{1}{7} a^{7} + \frac{3}{7} a^{6} - \frac{2}{7} a^{5} - \frac{3}{7} a^{4} + \frac{3}{7} a^{3} - \frac{3}{7} a^{2} + \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{14} + \frac{2}{7} a^{10} - \frac{3}{7} a^{9} + \frac{1}{7} a^{8} - \frac{1}{7} a^{7} - \frac{2}{7} a^{6} - \frac{1}{7} a^{5} - \frac{1}{7} a^{4} - \frac{2}{7} a^{3} - \frac{3}{7} a^{2} + \frac{2}{7} a - \frac{3}{7}$, $\frac{1}{7} a^{15} + \frac{2}{7} a^{11} - \frac{3}{7} a^{10} + \frac{1}{7} a^{9} - \frac{1}{7} a^{8} - \frac{2}{7} a^{7} - \frac{1}{7} a^{6} - \frac{1}{7} a^{5} - \frac{2}{7} a^{4} - \frac{3}{7} a^{3} + \frac{2}{7} a^{2} - \frac{3}{7} a$, $\frac{1}{7} a^{16} + \frac{2}{7} a^{11} - \frac{2}{7} a^{10} + \frac{2}{7} a^{9} - \frac{1}{7} a^{8} - \frac{3}{7} a^{7} - \frac{1}{7} a^{6} - \frac{1}{7} a^{5} + \frac{2}{7} a^{4} - \frac{1}{7} a^{3} + \frac{2}{7} a^{2} - \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{1417095150755041328753} a^{17} - \frac{12802585925733274598}{1417095150755041328753} a^{16} - \frac{23144714927346697938}{1417095150755041328753} a^{15} - \frac{42085709428907793045}{1417095150755041328753} a^{14} + \frac{30643100542639940581}{1417095150755041328753} a^{13} - \frac{33816425367433194947}{1417095150755041328753} a^{12} - \frac{252630441385702551673}{1417095150755041328753} a^{11} - \frac{563760062195415975086}{1417095150755041328753} a^{10} + \frac{514463754784328171429}{1417095150755041328753} a^{9} + \frac{23550982523116415294}{1417095150755041328753} a^{8} + \frac{307201710777432404403}{1417095150755041328753} a^{7} - \frac{224508247208288533904}{1417095150755041328753} a^{6} - \frac{44128193481813924628}{109007319288849332981} a^{5} - \frac{497988594411436833896}{1417095150755041328753} a^{4} + \frac{141259655793995241824}{1417095150755041328753} a^{3} - \frac{509102131994769415828}{1417095150755041328753} a^{2} - \frac{628092300736300593392}{1417095150755041328753} a + \frac{309935931367630385492}{1417095150755041328753}$

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:  \( 191842303.338 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T400:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2592
The 56 conjugacy class representatives for t18n400 are not computed
Character table for t18n400 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.4.199283.1, 9.9.26552265046321.1

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 $18$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ $18$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
$83$83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.6.5.2$x^{6} + 249$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.4.2.1$x^{4} + 6335 x^{2} + 10614564$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$