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Properties

Label 18.2.65683413076...0496.2
Degree $18$
Signature $[2, 8]$
Discriminant $2^{12}\cdot 7^{12}\cdot 41^{5}$
Root discriminant $16.30$
Ramified primes $2, 7, 41$
Class number $1$
Class group Trivial
Galois group 18T839

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, -4, -5, -2, -13, -3, -34, 2, -30, 3, -10, -11, 9, -6, 2, 1, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + x^16 + 2*x^15 - 6*x^14 + 9*x^13 - 11*x^12 - 10*x^11 + 3*x^10 - 30*x^9 + 2*x^8 - 34*x^7 - 3*x^6 - 13*x^5 - 2*x^4 - 5*x^3 - 4*x^2 - 2*x - 1)
 
gp: K = bnfinit(x^18 - x^17 + x^16 + 2*x^15 - 6*x^14 + 9*x^13 - 11*x^12 - 10*x^11 + 3*x^10 - 30*x^9 + 2*x^8 - 34*x^7 - 3*x^6 - 13*x^5 - 2*x^4 - 5*x^3 - 4*x^2 - 2*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + x^{16} + 2 x^{15} - 6 x^{14} + 9 x^{13} - 11 x^{12} - 10 x^{11} + 3 x^{10} - 30 x^{9} + 2 x^{8} - 34 x^{7} - 3 x^{6} - 13 x^{5} - 2 x^{4} - 5 x^{3} - 4 x^{2} - 2 x - 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(6568341307628680810496=2^{12}\cdot 7^{12}\cdot 41^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.30$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{32} a^{16} - \frac{11}{32} a^{15} - \frac{5}{16} a^{14} - \frac{7}{32} a^{13} - \frac{3}{16} a^{12} - \frac{3}{8} a^{11} + \frac{3}{32} a^{10} + \frac{1}{8} a^{9} - \frac{1}{2} a^{8} - \frac{1}{16} a^{7} + \frac{3}{16} a^{6} - \frac{3}{8} a^{5} - \frac{1}{32} a^{4} + \frac{9}{32} a^{3} - \frac{3}{32} a^{2} - \frac{1}{4} a - \frac{9}{32}$, $\frac{1}{9460736} a^{17} + \frac{45045}{4730368} a^{16} - \frac{1013297}{9460736} a^{15} - \frac{2298361}{9460736} a^{14} - \frac{3972761}{9460736} a^{13} - \frac{453813}{4730368} a^{12} + \frac{207271}{9460736} a^{11} - \frac{2241213}{9460736} a^{10} - \frac{523167}{2365184} a^{9} + \frac{1497095}{4730368} a^{8} - \frac{1098569}{2365184} a^{7} - \frac{110615}{4730368} a^{6} + \frac{2938819}{9460736} a^{5} + \frac{611389}{2365184} a^{4} + \frac{482813}{4730368} a^{3} + \frac{2744441}{9460736} a^{2} + \frac{2559503}{9460736} a + \frac{1666243}{9460736}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
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:  \( 4502.51392554 \)
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 $18$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ $18$ $18$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.12.11$x^{12} - 6 x^{10} - 73 x^{8} + 140 x^{6} + 79 x^{4} - 6 x^{2} + 57$$2$$6$$12$$A_4 \times C_2$$[2, 2]^{6}$
7Data not computed
41Data not computed