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Properties

Label 18.18.2559928162...3761.1
Degree $18$
Signature $[18, 0]$
Discriminant $19^{16}\cdot 31^{6}$
Root discriminant $43.03$
Ramified primes $19, 31$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2^2 : C_9$ (as 18T7)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 11, -275, 746, 2915, -7039, -15005, 10625, 21199, -8725, -12716, 4963, 3387, -1586, -296, 214, -9, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 - 9*x^16 + 214*x^15 - 296*x^14 - 1586*x^13 + 3387*x^12 + 4963*x^11 - 12716*x^10 - 8725*x^9 + 21199*x^8 + 10625*x^7 - 15005*x^6 - 7039*x^5 + 2915*x^4 + 746*x^3 - 275*x^2 + 11*x + 1)
 
gp: K = bnfinit(x^18 - 8*x^17 - 9*x^16 + 214*x^15 - 296*x^14 - 1586*x^13 + 3387*x^12 + 4963*x^11 - 12716*x^10 - 8725*x^9 + 21199*x^8 + 10625*x^7 - 15005*x^6 - 7039*x^5 + 2915*x^4 + 746*x^3 - 275*x^2 + 11*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} - 9 x^{16} + 214 x^{15} - 296 x^{14} - 1586 x^{13} + 3387 x^{12} + 4963 x^{11} - 12716 x^{10} - 8725 x^{9} + 21199 x^{8} + 10625 x^{7} - 15005 x^{6} - 7039 x^{5} + 2915 x^{4} + 746 x^{3} - 275 x^{2} + 11 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(255992816294107128730405733761=19^{16}\cdot 31^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.03$
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:  $19, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{447266660880125363609} a^{17} - \frac{72796902926401755909}{447266660880125363609} a^{16} + \frac{114853932345612552996}{447266660880125363609} a^{15} - \frac{3980209255442830216}{447266660880125363609} a^{14} + \frac{155976394723691982438}{447266660880125363609} a^{13} - \frac{180272825431091243324}{447266660880125363609} a^{12} - \frac{108475315386661227992}{447266660880125363609} a^{11} + \frac{49878995507626557885}{447266660880125363609} a^{10} + \frac{122069296680927554522}{447266660880125363609} a^{9} - \frac{174060156779372880941}{447266660880125363609} a^{8} - \frac{152120390728650530744}{447266660880125363609} a^{7} + \frac{137682050169567905042}{447266660880125363609} a^{6} + \frac{52862409659073502506}{447266660880125363609} a^{5} - \frac{4512258652208230915}{447266660880125363609} a^{4} - \frac{186593153465168509415}{447266660880125363609} a^{3} + \frac{76627776866245086444}{447266660880125363609} a^{2} - \frac{49073352783885796216}{447266660880125363609} a + \frac{1842346883182197381}{447266660880125363609}$

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:  \( 618241654.691 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:C_9$ (as 18T7):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 12 conjugacy class representatives for $C_2^2 : C_9$
Character table for $C_2^2 : C_9$

Intermediate fields

3.3.361.1, 6.6.125238481.1, \(\Q(\zeta_{19})^+\)

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

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$19$19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.8$x^{9} - 19$$9$$1$$8$$C_9$$[\ ]_{9}$
$31$31.6.3.2$x^{6} - 961 x^{2} + 268119$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
31.6.0.1$x^{6} - 2 x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
31.6.3.1$x^{6} - 62 x^{4} + 961 x^{2} - 2413071$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$