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Properties

Label 18.18.4120062536...9969.1
Degree $18$
Signature $[18, 0]$
Discriminant $7^{12}\cdot 19^{6}\cdot 293^{6}$
Root discriminant $64.85$
Ramified primes $7, 19, 293$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times A_5$ (as 18T90)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-127, -66, 3859, -1988, -25783, 15806, 55351, -22763, -52189, 11970, 23869, -2834, -5681, 300, 708, -11, -43, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 43*x^16 - 11*x^15 + 708*x^14 + 300*x^13 - 5681*x^12 - 2834*x^11 + 23869*x^10 + 11970*x^9 - 52189*x^8 - 22763*x^7 + 55351*x^6 + 15806*x^5 - 25783*x^4 - 1988*x^3 + 3859*x^2 - 66*x - 127)
 
gp: K = bnfinit(x^18 - 43*x^16 - 11*x^15 + 708*x^14 + 300*x^13 - 5681*x^12 - 2834*x^11 + 23869*x^10 + 11970*x^9 - 52189*x^8 - 22763*x^7 + 55351*x^6 + 15806*x^5 - 25783*x^4 - 1988*x^3 + 3859*x^2 - 66*x - 127, 1)
 

Normalized defining polynomial

\( x^{18} - 43 x^{16} - 11 x^{15} + 708 x^{14} + 300 x^{13} - 5681 x^{12} - 2834 x^{11} + 23869 x^{10} + 11970 x^{9} - 52189 x^{8} - 22763 x^{7} + 55351 x^{6} + 15806 x^{5} - 25783 x^{4} - 1988 x^{3} + 3859 x^{2} - 66 x - 127 \)

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:  \(412006253628151171906676331799969=7^{12}\cdot 19^{6}\cdot 293^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.85$
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:  $7, 19, 293$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{293} a^{16} + \frac{3}{293} a^{15} + \frac{37}{293} a^{14} + \frac{20}{293} a^{13} - \frac{121}{293} a^{12} - \frac{108}{293} a^{11} + \frac{54}{293} a^{10} - \frac{85}{293} a^{9} - \frac{94}{293} a^{8} + \frac{86}{293} a^{7} - \frac{5}{293} a^{6} + \frac{29}{293} a^{5} - \frac{1}{293} a^{4} - \frac{11}{293} a^{3} - \frac{103}{293} a^{2} + \frac{145}{293} a - \frac{89}{293}$, $\frac{1}{8792960650605717334709267} a^{17} + \frac{2119371120062038133574}{8792960650605717334709267} a^{16} + \frac{4021595456866379396320028}{8792960650605717334709267} a^{15} + \frac{2283122048923623961821917}{8792960650605717334709267} a^{14} - \frac{48710276998951547105766}{123844516205714328657877} a^{13} + \frac{500343733723836539897480}{8792960650605717334709267} a^{12} + \frac{55918657985075040138342}{8792960650605717334709267} a^{11} + \frac{2049461435132157289442697}{8792960650605717334709267} a^{10} + \frac{3258648147892775317799473}{8792960650605717334709267} a^{9} + \frac{1159723576896911363957267}{8792960650605717334709267} a^{8} - \frac{922554433064624050253359}{8792960650605717334709267} a^{7} + \frac{974656792184836504867122}{8792960650605717334709267} a^{6} - \frac{1173058658021194126945088}{8792960650605717334709267} a^{5} - \frac{269971881700531055190121}{8792960650605717334709267} a^{4} + \frac{57494290201610365246332}{123844516205714328657877} a^{3} + \frac{54881093938959152980985}{123844516205714328657877} a^{2} + \frac{2243198697620631813223864}{8792960650605717334709267} a + \frac{501011691090339444238184}{8792960650605717334709267}$

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:  \( 37446504667.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times A_5$ (as 18T90):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 180
The 15 conjugacy class representatives for $C_3\times A_5$
Character table for $C_3\times A_5$

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.6.30991489.1

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

Sibling fields

Degree 15 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 $15{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }$ $15{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ $15{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ $15{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ $15{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$19$19.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
19.6.3.1$x^{6} - 38 x^{4} + 361 x^{2} - 109744$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
19.6.3.2$x^{6} - 361 x^{2} + 27436$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
293Data not computed