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Properties

Label 18.6.26903925996...1616.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{24}\cdot 7^{12}\cdot 41^{5}$
Root discriminant $25.87$
Ramified primes $2, 7, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T839

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-41, 0, 222, 0, -230, 0, 41, 0, -169, 0, -197, 0, 11, 0, -10, 0, -5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^16 - 10*x^14 + 11*x^12 - 197*x^10 - 169*x^8 + 41*x^6 - 230*x^4 + 222*x^2 - 41)
 
gp: K = bnfinit(x^18 - 5*x^16 - 10*x^14 + 11*x^12 - 197*x^10 - 169*x^8 + 41*x^6 - 230*x^4 + 222*x^2 - 41, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{16} - 10 x^{14} + 11 x^{12} - 197 x^{10} - 169 x^{8} + 41 x^{6} - 230 x^{4} + 222 x^{2} - 41 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(26903925996047076599791616=2^{24}\cdot 7^{12}\cdot 41^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.87$
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:  $2, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} + \frac{1}{4} a^{8} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} + \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{4} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} + \frac{1}{4}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{12} - \frac{1}{2} a^{11} + \frac{1}{8} a^{10} - \frac{1}{4} a^{9} + \frac{3}{8} a^{8} + \frac{1}{4} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{8} a^{2} - \frac{3}{8} a + \frac{3}{8}$, $\frac{1}{204444229928} a^{16} + \frac{5392339441}{51111057482} a^{14} - \frac{1}{8} a^{13} + \frac{359125297}{51111057482} a^{12} + \frac{1}{8} a^{11} + \frac{25786178747}{102222114964} a^{10} + \frac{3}{8} a^{9} + \frac{17240389505}{102222114964} a^{8} - \frac{1}{8} a^{7} + \frac{9940915081}{25555528741} a^{6} - \frac{1}{8} a^{5} - \frac{16345402683}{51111057482} a^{4} + \frac{1}{8} a^{3} - \frac{18652283411}{204444229928} a^{2} + \frac{3}{8} a + \frac{7716915111}{51111057482}$, $\frac{1}{204444229928} a^{17} - \frac{3986170977}{204444229928} a^{15} - \frac{1}{8} a^{14} + \frac{359125297}{51111057482} a^{13} - \frac{25324878735}{102222114964} a^{11} - \frac{1}{2} a^{10} + \frac{21397959123}{51111057482} a^{9} + \frac{1}{4} a^{8} + \frac{14208131583}{102222114964} a^{7} - \frac{1}{4} a^{6} - \frac{16345402683}{51111057482} a^{5} + \frac{83569831553}{204444229928} a^{3} - \frac{1}{2} a^{2} - \frac{96909983261}{204444229928} a + \frac{3}{8}$

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:  $11$
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:  \( 805358.903198 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T839:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 165888
The 192 conjugacy class representatives for t18n839 are not computed
Character table for t18n839 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.5.12657150016.1

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 R $18$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R $18$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ $18$ $18$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$2$2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.12.18.48$x^{12} - 4 x^{11} - 4 x^{10} + 8 x^{9} - 4 x^{8} - 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{2} + 8$$4$$3$$18$$A_4\times C_2$$[2, 2]^{6}$
7Data not computed
41Data not computed