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Properties

Label 18.0.56773923432...5123.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 19^{16}$
Root discriminant $23.73$
Ramified primes $3, 19$
Class number $9$
Class group $[9]$
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![1, -5, 35, 10, 185, -71, 431, -77, 494, -118, 361, -63, 162, -29, 50, -6, 9, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 9*x^16 - 6*x^15 + 50*x^14 - 29*x^13 + 162*x^12 - 63*x^11 + 361*x^10 - 118*x^9 + 494*x^8 - 77*x^7 + 431*x^6 - 71*x^5 + 185*x^4 + 10*x^3 + 35*x^2 - 5*x + 1)
 
gp: K = bnfinit(x^18 - x^17 + 9*x^16 - 6*x^15 + 50*x^14 - 29*x^13 + 162*x^12 - 63*x^11 + 361*x^10 - 118*x^9 + 494*x^8 - 77*x^7 + 431*x^6 - 71*x^5 + 185*x^4 + 10*x^3 + 35*x^2 - 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 9 x^{16} - 6 x^{15} + 50 x^{14} - 29 x^{13} + 162 x^{12} - 63 x^{11} + 361 x^{10} - 118 x^{9} + 494 x^{8} - 77 x^{7} + 431 x^{6} - 71 x^{5} + 185 x^{4} + 10 x^{3} + 35 x^{2} - 5 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-5677392343251487443465123=-\,3^{9}\cdot 19^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.73$
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:  $3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(57=3\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{57}(1,·)$, $\chi_{57}(4,·)$, $\chi_{57}(5,·)$, $\chi_{57}(7,·)$, $\chi_{57}(11,·)$, $\chi_{57}(16,·)$, $\chi_{57}(17,·)$, $\chi_{57}(20,·)$, $\chi_{57}(23,·)$, $\chi_{57}(25,·)$, $\chi_{57}(26,·)$, $\chi_{57}(28,·)$, $\chi_{57}(35,·)$, $\chi_{57}(43,·)$, $\chi_{57}(44,·)$, $\chi_{57}(47,·)$, $\chi_{57}(49,·)$, $\chi_{57}(55,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{37} a^{16} - \frac{4}{37} a^{15} - \frac{12}{37} a^{14} + \frac{14}{37} a^{13} - \frac{3}{37} a^{12} - \frac{1}{37} a^{11} + \frac{5}{37} a^{10} - \frac{8}{37} a^{9} - \frac{2}{37} a^{8} + \frac{4}{37} a^{7} - \frac{7}{37} a^{6} - \frac{3}{37} a^{5} + \frac{5}{37} a^{4} + \frac{13}{37} a^{3} + \frac{18}{37} a^{2} + \frac{8}{37} a + \frac{9}{37}$, $\frac{1}{154764745241131} a^{17} + \frac{704294953363}{154764745241131} a^{16} + \frac{17035333877813}{154764745241131} a^{15} - \frac{21931839203463}{154764745241131} a^{14} - \frac{755893282713}{4182830952463} a^{13} - \frac{68703429952733}{154764745241131} a^{12} + \frac{2865531738805}{154764745241131} a^{11} + \frac{41922303650527}{154764745241131} a^{10} - \frac{64278683926278}{154764745241131} a^{9} - \frac{32436830274692}{154764745241131} a^{8} - \frac{63764950471438}{154764745241131} a^{7} - \frac{75054966409369}{154764745241131} a^{6} - \frac{54214255146582}{154764745241131} a^{5} - \frac{16333724524675}{154764745241131} a^{4} + \frac{56710263591819}{154764745241131} a^{3} + \frac{62141365181579}{154764745241131} a^{2} - \frac{1352008403848}{154764745241131} a - \frac{47076200659874}{154764745241131}$

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

Class group and class number

$C_{9}$, which has order $9$

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:  \( \frac{28416407390445}{154764745241131} a^{17} - \frac{26530391809334}{154764745241131} a^{16} + \frac{251486650478394}{154764745241131} a^{15} - \frac{150560620146021}{154764745241131} a^{14} + \frac{1388512765129213}{154764745241131} a^{13} - \frac{711364544797171}{154764745241131} a^{12} + \frac{4434676100278545}{154764745241131} a^{11} - \frac{1392636818955699}{154764745241131} a^{10} + \frac{9781341697275842}{154764745241131} a^{9} - \frac{2457651917355293}{154764745241131} a^{8} + \frac{13051931144375194}{154764745241131} a^{7} - \frac{819941733363259}{154764745241131} a^{6} + \frac{11130335830242394}{154764745241131} a^{5} - \frac{838871219953438}{154764745241131} a^{4} + \frac{4373617825317358}{154764745241131} a^{3} + \frac{1073864914854833}{154764745241131} a^{2} + \frac{763954466829073}{154764745241131} a + \frac{47644201890144}{154764745241131} \) (order $6$)
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:  \( 22305.8950792 \)
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

\(\Q(\sqrt{-3}) \), 3.3.361.1, 6.0.3518667.1, \(\Q(\zeta_{19})^+\)

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 $18$ R $18$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ $18$ R $18$ $18$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ $18$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ $18$ $18$

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
3Data not computed
$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}$