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Properties

Label 18.0.17941796507...4379.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,19^{17}\cdot 41^{9}$
Root discriminant $103.30$
Ramified primes $19, 41$
Class number $934990$ (GRH)
Class group $[934990]$ (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![62837670391, -43837670391, 43837670391, -15337670391, 15337670391, -2797670391, 2797670391, -289670391, 289670391, -17970391, 17970391, -680391, 680391, -15391, 15391, -191, 191, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 191*x^16 - 191*x^15 + 15391*x^14 - 15391*x^13 + 680391*x^12 - 680391*x^11 + 17970391*x^10 - 17970391*x^9 + 289670391*x^8 - 289670391*x^7 + 2797670391*x^6 - 2797670391*x^5 + 15337670391*x^4 - 15337670391*x^3 + 43837670391*x^2 - 43837670391*x + 62837670391)
 
gp: K = bnfinit(x^18 - x^17 + 191*x^16 - 191*x^15 + 15391*x^14 - 15391*x^13 + 680391*x^12 - 680391*x^11 + 17970391*x^10 - 17970391*x^9 + 289670391*x^8 - 289670391*x^7 + 2797670391*x^6 - 2797670391*x^5 + 15337670391*x^4 - 15337670391*x^3 + 43837670391*x^2 - 43837670391*x + 62837670391, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 191 x^{16} - 191 x^{15} + 15391 x^{14} - 15391 x^{13} + 680391 x^{12} - 680391 x^{11} + 17970391 x^{10} - 17970391 x^{9} + 289670391 x^{8} - 289670391 x^{7} + 2797670391 x^{6} - 2797670391 x^{5} + 15337670391 x^{4} - 15337670391 x^{3} + 43837670391 x^{2} - 43837670391 x + 62837670391 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1794179650728830182319895880000714379=-\,19^{17}\cdot 41^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $103.30$
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:  $19, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(779=19\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{779}(1,·)$, $\chi_{779}(450,·)$, $\chi_{779}(329,·)$, $\chi_{779}(778,·)$, $\chi_{779}(204,·)$, $\chi_{779}(206,·)$, $\chi_{779}(657,·)$, $\chi_{779}(83,·)$, $\chi_{779}(409,·)$, $\chi_{779}(737,·)$, $\chi_{779}(739,·)$, $\chi_{779}(40,·)$, $\chi_{779}(42,·)$, $\chi_{779}(370,·)$, $\chi_{779}(696,·)$, $\chi_{779}(122,·)$, $\chi_{779}(573,·)$, $\chi_{779}(575,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{9863177171} a^{10} + \frac{3741615072}{9863177171} a^{9} + \frac{100}{9863177171} a^{8} + \frac{1397332666}{9863177171} a^{7} + \frac{3500}{9863177171} a^{6} + \frac{2467271296}{9863177171} a^{5} + \frac{50000}{9863177171} a^{4} - \frac{4367223151}{9863177171} a^{3} + \frac{250000}{9863177171} a^{2} - \frac{3238492282}{9863177171} a + \frac{200000}{9863177171}$, $\frac{1}{9863177171} a^{11} + \frac{110}{9863177171} a^{9} + \frac{2036557964}{9863177171} a^{8} + \frac{4400}{9863177171} a^{7} - \frac{4749374787}{9863177171} a^{6} + \frac{77000}{9863177171} a^{5} - \frac{376243623}{9863177171} a^{4} + \frac{550000}{9863177171} a^{3} - \frac{3009948984}{9863177171} a^{2} + \frac{1100000}{9863177171} a - \frac{3762436230}{9863177171}$, $\frac{1}{9863177171} a^{12} + \frac{4712341226}{9863177171} a^{9} - \frac{6600}{9863177171} a^{8} - \frac{645133311}{9863177171} a^{7} - \frac{308000}{9863177171} a^{6} + \frac{4392874605}{9863177171} a^{5} - \frac{4950000}{9863177171} a^{4} + \frac{3952093418}{9863177171} a^{3} - \frac{26400000}{9863177171} a^{2} - \frac{2602663366}{9863177171} a - \frac{22000000}{9863177171}$, $\frac{1}{9863177171} a^{13} - \frac{7800}{9863177171} a^{9} + \frac{1553248297}{9863177171} a^{8} - \frac{416000}{9863177171} a^{7} + \frac{2430813517}{9863177171} a^{6} - \frac{8190000}{9863177171} a^{5} - \frac{1532945734}{9863177171} a^{4} - \frac{62400000}{9863177171} a^{3} - \frac{438327613}{9863177171} a^{2} - \frac{130000000}{9863177171} a - \frac{2213802266}{9863177171}$, $\frac{1}{9863177171} a^{14} + \frac{1009560908}{9863177171} a^{9} + \frac{364000}{9863177171} a^{8} + \frac{2814834362}{9863177171} a^{7} + \frac{19110000}{9863177171} a^{6} + \frac{124502445}{9863177171} a^{5} + \frac{327600000}{9863177171} a^{4} + \frac{2635043221}{9863177171} a^{3} + \frac{1820000000}{9863177171} a^{2} - \frac{2856866935}{9863177171} a + \frac{1560000000}{9863177171}$, $\frac{1}{9863177171} a^{15} + \frac{455000}{9863177171} a^{9} + \frac{490515272}{9863177171} a^{8} + \frac{27300000}{9863177171} a^{7} - \frac{2321248337}{9863177171} a^{6} + \frac{573300000}{9863177171} a^{5} + \frac{4330404399}{9863177171} a^{4} + \frac{4550000000}{9863177171} a^{3} - \frac{4243238216}{9863177171} a^{2} - \frac{113177171}{9863177171} a - \frac{3081732459}{9863177171}$, $\frac{1}{9863177171} a^{16} - \frac{671644273}{9863177171} a^{9} - \frac{18200000}{9863177171} a^{8} + \frac{1579341494}{9863177171} a^{7} - \frac{1019200000}{9863177171} a^{6} + \frac{2989973277}{9863177171} a^{5} + \frac{1526354342}{9863177171} a^{4} - \frac{2698288731}{9863177171} a^{3} + \frac{4494948881}{9863177171} a^{2} + \frac{1553115996}{9863177171} a - \frac{2231405461}{9863177171}$, $\frac{1}{9863177171} a^{17} - \frac{23800000}{9863177171} a^{9} - \frac{298471403}{9863177171} a^{8} - \frac{1523200000}{9863177171} a^{7} - \frac{3554415092}{9863177171} a^{6} - \frac{3730468487}{9863177171} a^{5} - \frac{4602905986}{9863177171} a^{4} + \frac{4168960788}{9863177171} a^{3} + \frac{1893206892}{9863177171} a^{2} - \frac{3209369740}{9863177171} a + \frac{2244708151}{9863177171}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{934990}$, which has order $934990$ (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:  $8$
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:  \( 22305.895079162343 \) (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

\(\Q(\sqrt{-779}) \), 3.3.361.1, 6.0.170655219179.1, \(\Q(\zeta_{19})^+\)

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 $18$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ $18$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ $18$

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
19Data not computed
41Data not computed