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Properties

Label 18.6.20946604894...7089.1
Degree $18$
Signature $[6, 6]$
Discriminant $7^{12}\cdot 73^{6}$
Root discriminant $15.29$
Ramified primes $7, 73$
Class number $1$
Class group Trivial
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![1, -11, 52, -135, 194, -93, -214, 554, -674, 521, -267, 103, -58, 44, -11, -17, 18, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 18*x^16 - 17*x^15 - 11*x^14 + 44*x^13 - 58*x^12 + 103*x^11 - 267*x^10 + 521*x^9 - 674*x^8 + 554*x^7 - 214*x^6 - 93*x^5 + 194*x^4 - 135*x^3 + 52*x^2 - 11*x + 1)
 
gp: K = bnfinit(x^18 - 7*x^17 + 18*x^16 - 17*x^15 - 11*x^14 + 44*x^13 - 58*x^12 + 103*x^11 - 267*x^10 + 521*x^9 - 674*x^8 + 554*x^7 - 214*x^6 - 93*x^5 + 194*x^4 - 135*x^3 + 52*x^2 - 11*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 18 x^{16} - 17 x^{15} - 11 x^{14} + 44 x^{13} - 58 x^{12} + 103 x^{11} - 267 x^{10} + 521 x^{9} - 674 x^{8} + 554 x^{7} - 214 x^{6} - 93 x^{5} + 194 x^{4} - 135 x^{3} + 52 x^{2} - 11 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(2094660489407173427089=7^{12}\cdot 73^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.29$
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:  $7, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{329} a^{15} + \frac{150}{329} a^{14} + \frac{55}{329} a^{13} - \frac{1}{47} a^{12} + \frac{20}{47} a^{11} + \frac{17}{329} a^{10} + \frac{20}{47} a^{9} - \frac{86}{329} a^{8} - \frac{143}{329} a^{7} + \frac{57}{329} a^{6} + \frac{115}{329} a^{5} + \frac{104}{329} a^{4} - \frac{8}{329} a^{3} - \frac{43}{329} a^{2} + \frac{164}{329} a + \frac{1}{329}$, $\frac{1}{987} a^{16} - \frac{73}{987} a^{14} + \frac{99}{329} a^{13} + \frac{29}{141} a^{12} + \frac{73}{987} a^{11} - \frac{107}{987} a^{10} - \frac{359}{987} a^{9} + \frac{85}{329} a^{8} + \frac{451}{987} a^{7} + \frac{17}{141} a^{6} + \frac{97}{329} a^{5} - \frac{158}{329} a^{4} - \frac{53}{329} a^{3} - \frac{295}{987} a^{2} - \frac{253}{987} a - \frac{479}{987}$, $\frac{1}{2961} a^{17} + \frac{1}{2961} a^{16} - \frac{1}{2961} a^{15} + \frac{1154}{2961} a^{14} - \frac{1462}{2961} a^{13} + \frac{253}{987} a^{12} + \frac{1163}{2961} a^{11} + \frac{758}{2961} a^{10} - \frac{881}{2961} a^{9} - \frac{551}{2961} a^{8} - \frac{281}{987} a^{7} - \frac{421}{2961} a^{6} + \frac{132}{329} a^{5} + \frac{311}{987} a^{4} + \frac{944}{2961} a^{3} - \frac{683}{2961} a^{2} - \frac{256}{987} a - \frac{1394}{2961}$

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:  $11$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 4120.17595553 \)
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

\(\Q(\zeta_{7})^+\), 6.2.12794929.1

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 ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ $15{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ $15{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$7.9.6.1$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}$
$73$73.6.3.2$x^{6} - 5329 x^{2} + 5446238$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
73.6.0.1$x^{6} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
73.6.3.1$x^{6} - 146 x^{4} + 5329 x^{2} - 76247332$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$