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Properties

Label 18.18.1766000400...6521.1
Degree $18$
Signature $[18, 0]$
Discriminant $7^{10}\cdot 569^{9}$
Root discriminant $70.32$
Ramified primes $7, 569$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $He_3:C_2$ (as 18T22)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![200, -460, -2996, 7729, 9810, -30769, -9734, 49453, -418, -37698, 5233, 14323, -2617, -2716, 515, 232, -42, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 42*x^16 + 232*x^15 + 515*x^14 - 2716*x^13 - 2617*x^12 + 14323*x^11 + 5233*x^10 - 37698*x^9 - 418*x^8 + 49453*x^7 - 9734*x^6 - 30769*x^5 + 9810*x^4 + 7729*x^3 - 2996*x^2 - 460*x + 200)
 
gp: K = bnfinit(x^18 - 6*x^17 - 42*x^16 + 232*x^15 + 515*x^14 - 2716*x^13 - 2617*x^12 + 14323*x^11 + 5233*x^10 - 37698*x^9 - 418*x^8 + 49453*x^7 - 9734*x^6 - 30769*x^5 + 9810*x^4 + 7729*x^3 - 2996*x^2 - 460*x + 200, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 42 x^{16} + 232 x^{15} + 515 x^{14} - 2716 x^{13} - 2617 x^{12} + 14323 x^{11} + 5233 x^{10} - 37698 x^{9} - 418 x^{8} + 49453 x^{7} - 9734 x^{6} - 30769 x^{5} + 9810 x^{4} + 7729 x^{3} - 2996 x^{2} - 460 x + 200 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1766000400360784868524139223996521=7^{10}\cdot 569^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.32$
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:  $7, 569$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{28} a^{15} + \frac{1}{14} a^{14} - \frac{3}{28} a^{13} + \frac{1}{28} a^{12} - \frac{3}{28} a^{11} + \frac{1}{14} a^{10} - \frac{1}{7} a^{9} - \frac{3}{28} a^{8} + \frac{3}{7} a^{7} - \frac{1}{4} a^{6} + \frac{1}{28} a^{5} - \frac{13}{28} a^{4} + \frac{3}{14} a^{3} + \frac{5}{14} a^{2} + \frac{5}{14} a + \frac{1}{7}$, $\frac{1}{28} a^{16} + \frac{1}{14} a^{12} - \frac{3}{14} a^{11} + \frac{3}{14} a^{10} - \frac{1}{14} a^{9} + \frac{1}{7} a^{8} + \frac{1}{7} a^{7} + \frac{2}{7} a^{6} + \frac{3}{14} a^{5} - \frac{5}{14} a^{4} - \frac{1}{14} a^{3} - \frac{3}{28} a^{2} + \frac{3}{7} a - \frac{2}{7}$, $\frac{1}{216105817320127097014540} a^{17} + \frac{1653643709488859491969}{216105817320127097014540} a^{16} + \frac{1223799030615913294019}{108052908660063548507270} a^{15} + \frac{4637066213443792327438}{54026454330031774253635} a^{14} + \frac{869016011362689832401}{10805290866006354850727} a^{13} - \frac{18500035001344208450883}{108052908660063548507270} a^{12} - \frac{10634379273448178709428}{54026454330031774253635} a^{11} - \frac{17303407478752257802921}{108052908660063548507270} a^{10} - \frac{4055220571940472919111}{108052908660063548507270} a^{9} + \frac{1282863422805559488401}{108052908660063548507270} a^{8} - \frac{3327699783655046559181}{7718064904290253464805} a^{7} - \frac{4386236236933528726683}{54026454330031774253635} a^{6} - \frac{5533814415585251590591}{15436129808580506929610} a^{5} - \frac{4452368859006397346851}{54026454330031774253635} a^{4} - \frac{20811935186951312015159}{43221163464025419402908} a^{3} - \frac{6169330695700304331673}{30872259617161013859220} a^{2} - \frac{433993790199767244072}{7718064904290253464805} a + \frac{2558735431954552526380}{10805290866006354850727}$

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:  $17$
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:  \( 283969563865 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$He_3:C_2$ (as 18T22):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $He_3:C_2$
Character table for $He_3:C_2$

Intermediate fields

\(\Q(\sqrt{569}) \), 6.6.9026780441.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 9 siblings: data not computed
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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{5}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{5}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
569Data not computed