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Properties

Label 18.0.60205903544...7879.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 13^{9}\cdot 19^{16}$
Root discriminant $85.55$
Ramified primes $3, 13, 19$
Class number $143164$ (GRH)
Class group $[143164]$ (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![3945435121, -1667313198, 2585263350, -962623287, 792133944, -258570274, 148181983, -42122997, 18591955, -4550063, 1618870, -334121, 97606, -16343, 3915, -490, 94, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 94*x^16 - 490*x^15 + 3915*x^14 - 16343*x^13 + 97606*x^12 - 334121*x^11 + 1618870*x^10 - 4550063*x^9 + 18591955*x^8 - 42122997*x^7 + 148181983*x^6 - 258570274*x^5 + 792133944*x^4 - 962623287*x^3 + 2585263350*x^2 - 1667313198*x + 3945435121)
 
gp: K = bnfinit(x^18 - 7*x^17 + 94*x^16 - 490*x^15 + 3915*x^14 - 16343*x^13 + 97606*x^12 - 334121*x^11 + 1618870*x^10 - 4550063*x^9 + 18591955*x^8 - 42122997*x^7 + 148181983*x^6 - 258570274*x^5 + 792133944*x^4 - 962623287*x^3 + 2585263350*x^2 - 1667313198*x + 3945435121, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 94 x^{16} - 490 x^{15} + 3915 x^{14} - 16343 x^{13} + 97606 x^{12} - 334121 x^{11} + 1618870 x^{10} - 4550063 x^{9} + 18591955 x^{8} - 42122997 x^{7} + 148181983 x^{6} - 258570274 x^{5} + 792133944 x^{4} - 962623287 x^{3} + 2585263350 x^{2} - 1667313198 x + 3945435121 \)

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:  \(-60205903544285399375543269800867879=-\,3^{9}\cdot 13^{9}\cdot 19^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $85.55$
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:  $3, 13, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(741=3\cdot 13\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{741}(1,·)$, $\chi_{741}(194,·)$, $\chi_{741}(196,·)$, $\chi_{741}(389,·)$, $\chi_{741}(391,·)$, $\chi_{741}(586,·)$, $\chi_{741}(77,·)$, $\chi_{741}(272,·)$, $\chi_{741}(467,·)$, $\chi_{741}(662,·)$, $\chi_{741}(157,·)$, $\chi_{741}(233,·)$, $\chi_{741}(235,·)$, $\chi_{741}(625,·)$, $\chi_{741}(118,·)$, $\chi_{741}(311,·)$, $\chi_{741}(313,·)$, $\chi_{741}(701,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{65139102827137114919668136370872913198024860018272269503803} a^{17} - \frac{14334585919848815946250427751163688237369402316875701817703}{65139102827137114919668136370872913198024860018272269503803} a^{16} - \frac{12029236950349289171039751219160566968899015375883384607820}{65139102827137114919668136370872913198024860018272269503803} a^{15} + \frac{16410518047786392340339823632708432877360326853230836002818}{65139102827137114919668136370872913198024860018272269503803} a^{14} - \frac{9860724844309768424715546152107559778489480223280271079150}{65139102827137114919668136370872913198024860018272269503803} a^{13} + \frac{6217551907800823596946989892524879364935991033878214593328}{65139102827137114919668136370872913198024860018272269503803} a^{12} - \frac{18717947527497194329808057400474890643607145572510067295172}{65139102827137114919668136370872913198024860018272269503803} a^{11} + \frac{14115141449879530269694384300680630921188410047191060909915}{65139102827137114919668136370872913198024860018272269503803} a^{10} - \frac{24102360655394613551382625322455791863093712112810318926957}{65139102827137114919668136370872913198024860018272269503803} a^{9} + \frac{20225593285442986762686972216460951977147187106284381330437}{65139102827137114919668136370872913198024860018272269503803} a^{8} + \frac{22851528691079563690694464526315896507349999593800618111933}{65139102827137114919668136370872913198024860018272269503803} a^{7} + \frac{3499177403145066409154739713361436591569727004940874912717}{65139102827137114919668136370872913198024860018272269503803} a^{6} + \frac{17099633599736936616907468564152821085473809722309944992606}{65139102827137114919668136370872913198024860018272269503803} a^{5} + \frac{24071023384238707204551328436115062244468487611794648700740}{65139102827137114919668136370872913198024860018272269503803} a^{4} + \frac{7632552736158727679783124157945879361511547325900026511550}{65139102827137114919668136370872913198024860018272269503803} a^{3} + \frac{16322245127347174639341344370327779337324761677217006540965}{65139102827137114919668136370872913198024860018272269503803} a^{2} - \frac{26622651015271455038360860005326918991715743359770774930346}{65139102827137114919668136370872913198024860018272269503803} a - \frac{4036486791657949813109130411971660922624433840907318529333}{65139102827137114919668136370872913198024860018272269503803}$

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

Class group and class number

$C_{143164}$, which has order $143164$ (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:  \( 22305.8950792 \) (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

\(\Q(\sqrt{-39}) \), 3.3.361.1, 6.0.7730511399.2, \(\Q(\zeta_{19})^+\)

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R $18$ R $18$ $18$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
3Data not computed
13Data not computed
19Data not computed