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Properties

Label 18.10.2415964169...8133.2
Degree $18$
Signature $[10, 4]$
Discriminant $3^{18}\cdot 7^{14}\cdot 13\cdot 29^{4}$
Root discriminant $33.21$
Ramified primes $3, 7, 13, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T766

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![71, 678, 2145, 1884, -2400, -3843, 1014, 1695, -1152, 256, 708, -444, 34, 105, -87, 11, 6, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 6*x^16 + 11*x^15 - 87*x^14 + 105*x^13 + 34*x^12 - 444*x^11 + 708*x^10 + 256*x^9 - 1152*x^8 + 1695*x^7 + 1014*x^6 - 3843*x^5 - 2400*x^4 + 1884*x^3 + 2145*x^2 + 678*x + 71)
 
gp: K = bnfinit(x^18 - 3*x^17 + 6*x^16 + 11*x^15 - 87*x^14 + 105*x^13 + 34*x^12 - 444*x^11 + 708*x^10 + 256*x^9 - 1152*x^8 + 1695*x^7 + 1014*x^6 - 3843*x^5 - 2400*x^4 + 1884*x^3 + 2145*x^2 + 678*x + 71, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 6 x^{16} + 11 x^{15} - 87 x^{14} + 105 x^{13} + 34 x^{12} - 444 x^{11} + 708 x^{10} + 256 x^{9} - 1152 x^{8} + 1695 x^{7} + 1014 x^{6} - 3843 x^{5} - 2400 x^{4} + 1884 x^{3} + 2145 x^{2} + 678 x + 71 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2415964169284813676020898133=3^{18}\cdot 7^{14}\cdot 13\cdot 29^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.21$
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:  $3, 7, 13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{359077701271306259063081} a^{17} + \frac{162228024240026935392143}{359077701271306259063081} a^{16} - \frac{61370219410529156494905}{359077701271306259063081} a^{15} - \frac{22348415075334658168412}{359077701271306259063081} a^{14} - \frac{69725008140171252234746}{359077701271306259063081} a^{13} - \frac{7177682960178943953382}{27621361636254327620237} a^{12} - \frac{116830623172997698831521}{359077701271306259063081} a^{11} + \frac{87858608884517974643223}{359077701271306259063081} a^{10} + \frac{77696337944887329718101}{359077701271306259063081} a^{9} + \frac{5987005128190353977764}{27621361636254327620237} a^{8} - \frac{104245005629758224174380}{359077701271306259063081} a^{7} + \frac{171816191403201985240898}{359077701271306259063081} a^{6} - \frac{135402082388056349570671}{359077701271306259063081} a^{5} + \frac{44827866372333019816554}{359077701271306259063081} a^{4} + \frac{26006906673965085454751}{359077701271306259063081} a^{3} + \frac{18353471213582948591514}{359077701271306259063081} a^{2} - \frac{22183644978247871303089}{359077701271306259063081} a + \frac{107873698775617267215692}{359077701271306259063081}$

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:  $13$
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:  \( 23352776.7116 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T766:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 144 conjugacy class representatives for t18n766 are not computed
Character table for t18n766 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.13632439166829.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ R $18$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ $18$

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
$3$3.9.9.2$x^{9} + 18 x^{3} + 27 x + 27$$3$$3$$9$$C_3^2 : S_3 $$[3/2, 3/2]_{2}^{3}$
3.9.9.2$x^{9} + 18 x^{3} + 27 x + 27$$3$$3$$9$$C_3^2 : S_3 $$[3/2, 3/2]_{2}^{3}$
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.6.4.1$x^{6} + 232 x^{3} + 22707$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$