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Properties

Label 18.10.1052263120...0561.1
Degree $18$
Signature $[10, 4]$
Discriminant $19^{16}\cdot 191^{2}$
Root discriminant $24.55$
Ramified primes $19, 191$
Class number $1$
Class group Trivial
Galois group 18T368

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, -90, 104, 402, -473, -880, 1641, -171, -1402, 1102, 9, -446, 260, -39, -32, 25, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 + 25*x^16 - 32*x^15 - 39*x^14 + 260*x^13 - 446*x^12 + 9*x^11 + 1102*x^10 - 1402*x^9 - 171*x^8 + 1641*x^7 - 880*x^6 - 473*x^5 + 402*x^4 + 104*x^3 - 90*x^2 - 3*x + 1)
 
gp: K = bnfinit(x^18 - 8*x^17 + 25*x^16 - 32*x^15 - 39*x^14 + 260*x^13 - 446*x^12 + 9*x^11 + 1102*x^10 - 1402*x^9 - 171*x^8 + 1641*x^7 - 880*x^6 - 473*x^5 + 402*x^4 + 104*x^3 - 90*x^2 - 3*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} + 25 x^{16} - 32 x^{15} - 39 x^{14} + 260 x^{13} - 446 x^{12} + 9 x^{11} + 1102 x^{10} - 1402 x^{9} - 171 x^{8} + 1641 x^{7} - 880 x^{6} - 473 x^{5} + 402 x^{4} + 104 x^{3} - 90 x^{2} - 3 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(10522631208360387818170561=19^{16}\cdot 191^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.55$
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:  $19, 191$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} - \frac{1}{7} a^{14} + \frac{3}{7} a^{13} - \frac{2}{7} a^{12} - \frac{1}{7} a^{11} - \frac{1}{7} a^{10} + \frac{1}{7} a^{9} + \frac{2}{7} a^{8} + \frac{1}{7} a^{7} - \frac{3}{7} a^{6} - \frac{2}{7} a^{5} + \frac{1}{7} a^{3} + \frac{1}{7} a^{2} + \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{16} + \frac{2}{7} a^{14} + \frac{1}{7} a^{13} - \frac{3}{7} a^{12} - \frac{2}{7} a^{11} + \frac{3}{7} a^{9} + \frac{3}{7} a^{8} - \frac{2}{7} a^{7} + \frac{2}{7} a^{6} - \frac{2}{7} a^{5} + \frac{1}{7} a^{4} + \frac{2}{7} a^{3} + \frac{3}{7} a^{2} + \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{295577217509} a^{17} - \frac{1312261454}{42225316787} a^{16} + \frac{8714521241}{295577217509} a^{15} - \frac{39491558095}{295577217509} a^{14} - \frac{137815613710}{295577217509} a^{13} + \frac{67953937272}{295577217509} a^{12} - \frac{66644459200}{295577217509} a^{11} - \frac{93721369932}{295577217509} a^{10} + \frac{3436572573}{42225316787} a^{9} - \frac{45477812661}{295577217509} a^{8} - \frac{71242234449}{295577217509} a^{7} - \frac{2409536869}{42225316787} a^{6} - \frac{10492820313}{42225316787} a^{5} - \frac{137421353991}{295577217509} a^{4} - \frac{17411008155}{42225316787} a^{3} + \frac{31518389117}{295577217509} a^{2} - \frac{92665485}{42225316787} a - \frac{111885272993}{295577217509}$

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:  $13$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 694739.500993 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T368:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 40 conjugacy class representatives for t18n368
Character table for t18n368 is not computed

Intermediate fields

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

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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}$
191Data not computed