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Properties

Label 18.18.3058776789...4013.1
Degree $18$
Signature $[18, 0]$
Discriminant $13^{9}\cdot 19^{16}$
Root discriminant $49.39$
Ramified primes $13, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-191, 1022, 2633, -25352, 41513, 27104, -112670, 38369, 80696, -48410, -22172, 18205, 2147, -3057, 67, 238, -23, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 - 23*x^16 + 238*x^15 + 67*x^14 - 3057*x^13 + 2147*x^12 + 18205*x^11 - 22172*x^10 - 48410*x^9 + 80696*x^8 + 38369*x^7 - 112670*x^6 + 27104*x^5 + 41513*x^4 - 25352*x^3 + 2633*x^2 + 1022*x - 191)
 
gp: K = bnfinit(x^18 - 7*x^17 - 23*x^16 + 238*x^15 + 67*x^14 - 3057*x^13 + 2147*x^12 + 18205*x^11 - 22172*x^10 - 48410*x^9 + 80696*x^8 + 38369*x^7 - 112670*x^6 + 27104*x^5 + 41513*x^4 - 25352*x^3 + 2633*x^2 + 1022*x - 191, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} - 23 x^{16} + 238 x^{15} + 67 x^{14} - 3057 x^{13} + 2147 x^{12} + 18205 x^{11} - 22172 x^{10} - 48410 x^{9} + 80696 x^{8} + 38369 x^{7} - 112670 x^{6} + 27104 x^{5} + 41513 x^{4} - 25352 x^{3} + 2633 x^{2} + 1022 x - 191 \)

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:  \(3058776789325072365774692364013=13^{9}\cdot 19^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.39$
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:  $13, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(247=13\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{247}(64,·)$, $\chi_{247}(1,·)$, $\chi_{247}(66,·)$, $\chi_{247}(131,·)$, $\chi_{247}(196,·)$, $\chi_{247}(194,·)$, $\chi_{247}(142,·)$, $\chi_{247}(77,·)$, $\chi_{247}(144,·)$, $\chi_{247}(92,·)$, $\chi_{247}(25,·)$, $\chi_{247}(207,·)$, $\chi_{247}(220,·)$, $\chi_{247}(157,·)$, $\chi_{247}(168,·)$, $\chi_{247}(233,·)$, $\chi_{247}(235,·)$, $\chi_{247}(118,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{3366969913669514036406540857231} a^{17} - \frac{180068769397593610820039000072}{3366969913669514036406540857231} a^{16} + \frac{1491751870375529560298285325374}{3366969913669514036406540857231} a^{15} + \frac{1178811404046750978242720418952}{3366969913669514036406540857231} a^{14} - \frac{1446408662700153273280612403511}{3366969913669514036406540857231} a^{13} - \frac{1543966397945289711687832132656}{3366969913669514036406540857231} a^{12} - \frac{1074983562553987161931164563216}{3366969913669514036406540857231} a^{11} + \frac{529292718475768992071479111013}{3366969913669514036406540857231} a^{10} - \frac{795037352408329159527720202760}{3366969913669514036406540857231} a^{9} + \frac{313638757703288982503117219574}{3366969913669514036406540857231} a^{8} - \frac{1669854595777211040527259636876}{3366969913669514036406540857231} a^{7} - \frac{1044200514158512633129783864587}{3366969913669514036406540857231} a^{6} - \frac{1419572466366230498130993927027}{3366969913669514036406540857231} a^{5} + \frac{657160163940232621875392990042}{3366969913669514036406540857231} a^{4} + \frac{1638268026873452266258344673171}{3366969913669514036406540857231} a^{3} + \frac{1020899281370359255486389092382}{3366969913669514036406540857231} a^{2} + \frac{810480533934175512929738485757}{3366969913669514036406540857231} a + \frac{978713262736902235816357094295}{3366969913669514036406540857231}$

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:  \( 1710187464.86 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{13}) \), 3.3.361.1, 6.6.286315237.1, \(\Q(\zeta_{19})^+\)

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $18$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ $18$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ $18$

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
13Data not computed
19Data not computed