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Properties

Label 18.0.50620901514...3407.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,19^{15}\cdot 37^{15}$
Root discriminant $235.76$
Ramified primes $19, 37$
Class number $15134336$ (GRH)
Class group $[2, 2, 2, 28, 67564]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2852917793, -452031866, -1350313403, -1564440675, 2209461811, -646274448, 152081560, -130864084, 43928444, -3107148, -88308, -130372, -47312, 39408, -3197, -786, 95, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 95*x^16 - 786*x^15 - 3197*x^14 + 39408*x^13 - 47312*x^12 - 130372*x^11 - 88308*x^10 - 3107148*x^9 + 43928444*x^8 - 130864084*x^7 + 152081560*x^6 - 646274448*x^5 + 2209461811*x^4 - 1564440675*x^3 - 1350313403*x^2 - 452031866*x + 2852917793)
 
gp: K = bnfinit(x^18 - x^17 + 95*x^16 - 786*x^15 - 3197*x^14 + 39408*x^13 - 47312*x^12 - 130372*x^11 - 88308*x^10 - 3107148*x^9 + 43928444*x^8 - 130864084*x^7 + 152081560*x^6 - 646274448*x^5 + 2209461811*x^4 - 1564440675*x^3 - 1350313403*x^2 - 452031866*x + 2852917793, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 95 x^{16} - 786 x^{15} - 3197 x^{14} + 39408 x^{13} - 47312 x^{12} - 130372 x^{11} - 88308 x^{10} - 3107148 x^{9} + 43928444 x^{8} - 130864084 x^{7} + 152081560 x^{6} - 646274448 x^{5} + 2209461811 x^{4} - 1564440675 x^{3} - 1350313403 x^{2} - 452031866 x + 2852917793 \)

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:  \(-5062090151414178056069202621780473318873407=-\,19^{15}\cdot 37^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $235.76$
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:  $19, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(703=19\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{703}(1,·)$, $\chi_{703}(581,·)$, $\chi_{703}(582,·)$, $\chi_{703}(455,·)$, $\chi_{703}(334,·)$, $\chi_{703}(343,·)$, $\chi_{703}(26,·)$, $\chi_{703}(27,·)$, $\chi_{703}(221,·)$, $\chi_{703}(482,·)$, $\chi_{703}(676,·)$, $\chi_{703}(677,·)$, $\chi_{703}(360,·)$, $\chi_{703}(369,·)$, $\chi_{703}(248,·)$, $\chi_{703}(121,·)$, $\chi_{703}(122,·)$, $\chi_{703}(702,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{22} a^{11} - \frac{1}{2} a^{4} + \frac{5}{11} a$, $\frac{1}{22} a^{12} - \frac{1}{2} a^{5} + \frac{5}{11} a^{2}$, $\frac{1}{140492} a^{13} + \frac{1143}{140492} a^{12} + \frac{2241}{140492} a^{11} + \frac{1821}{12772} a^{10} - \frac{83}{12772} a^{9} + \frac{3133}{12772} a^{8} - \frac{2501}{12772} a^{7} - \frac{195}{412} a^{6} - \frac{4059}{12772} a^{5} + \frac{5931}{12772} a^{4} + \frac{16059}{140492} a^{3} - \frac{4641}{140492} a^{2} - \frac{52665}{140492} a + \frac{25}{124}$, $\frac{1}{140492} a^{14} - \frac{366}{35123} a^{12} + \frac{49}{35123} a^{11} + \frac{175}{6386} a^{10} + \frac{553}{3193} a^{9} - \frac{487}{6386} a^{8} - \frac{486}{3193} a^{7} - \frac{1069}{3193} a^{6} - \frac{910}{3193} a^{5} + \frac{11712}{35123} a^{4} + \frac{2017}{6386} a^{3} - \frac{13692}{35123} a^{2} + \frac{5413}{70246} a + \frac{7}{124}$, $\frac{1}{140492} a^{15} + \frac{104}{35123} a^{12} + \frac{1135}{70246} a^{11} - \frac{298}{3193} a^{10} - \frac{288}{3193} a^{9} - \frac{95}{3193} a^{8} - \frac{44}{3193} a^{7} - \frac{631}{3193} a^{6} + \frac{2373}{35123} a^{5} + \frac{1029}{6386} a^{4} - \frac{149}{3193} a^{3} + \frac{4373}{35123} a^{2} - \frac{52925}{140492} a + \frac{5}{31}$, $\frac{1}{8957348444} a^{16} - \frac{6629}{4478674222} a^{15} - \frac{8325}{4478674222} a^{14} + \frac{14023}{4478674222} a^{13} + \frac{129969}{21741137} a^{12} + \frac{49710530}{2239337111} a^{11} + \frac{25253159}{203576101} a^{10} - \frac{5976932}{203576101} a^{9} + \frac{41756090}{203576101} a^{8} + \frac{22634486}{203576101} a^{7} - \frac{569263789}{4478674222} a^{6} + \frac{436743976}{2239337111} a^{5} + \frac{134091704}{2239337111} a^{4} - \frac{284682188}{2239337111} a^{3} - \frac{2913483197}{8957348444} a^{2} - \frac{1248619583}{4478674222} a - \frac{980359}{3952934}$, $\frac{1}{1597892901874407791907301042952928044355757188248840443171792} a^{17} + \frac{22713949645545410967907702699765027353222502376539}{798946450937203895953650521476464022177878594124420221585896} a^{16} + \frac{5479204549676496217565093863616806499296527860658580849}{1597892901874407791907301042952928044355757188248840443171792} a^{15} + \frac{3148352873140285281922426929692093446880944428417570225}{1597892901874407791907301042952928044355757188248840443171792} a^{14} + \frac{2669855239355933014514984333638691402084577954659716369}{798946450937203895953650521476464022177878594124420221585896} a^{13} - \frac{492349204051062041313727331200126204633715345867563659647}{25772466159264641804956468434724645876705761100787749083416} a^{12} + \frac{1516284122136187599467779050937947146064835331555405896235}{72631495539745808723059138316042183834352599465856383780536} a^{11} + \frac{6472386984877033061556280559453808683723664510734305757239}{72631495539745808723059138316042183834352599465856383780536} a^{10} + \frac{9843709252477433291958989457325130406258001167835126261463}{72631495539745808723059138316042183834352599465856383780536} a^{9} - \frac{3239204938623398412520422926052426376867583720561778717549}{72631495539745808723059138316042183834352599465856383780536} a^{8} + \frac{146188383956854770702104632557061247696349522136988625091433}{798946450937203895953650521476464022177878594124420221585896} a^{7} - \frac{916071295641881360954576264600008732120602860535806221085}{25772466159264641804956468434724645876705761100787749083416} a^{6} - \frac{272975390649305503720159270116059605272616199315776048392601}{798946450937203895953650521476464022177878594124420221585896} a^{5} + \frac{189812177803059554136736467353227788190017539252980192844441}{798946450937203895953650521476464022177878594124420221585896} a^{4} - \frac{744586306849544399160953187606330539119900640170313268191711}{1597892901874407791907301042952928044355757188248840443171792} a^{3} + \frac{38044167425474278787378969669998690740613002849176377926523}{399473225468601947976825260738232011088939297062210110792948} a^{2} - \frac{31597803487639945875838561468553858910726656250184374728725}{145262991079491617446118276632084367668705198931712767561072} a - \frac{41262923712131527105976950523269045893124495132140368337}{128210936522057914780333871696455752576085788995333422384}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{28}\times C_{67564}$, which has order $15134336$ (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:  \( 57298269.80370742 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-703}) \), 3.3.1369.1, 3.3.494209.1, 3.3.494209.2, 3.3.361.1, 6.0.475630201063.1, 6.0.171702502583743.2, 6.0.171702502583743.1, 6.0.125421842647.1, 9.9.120706859316371329.1

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{18}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$37$37.6.5.5$x^{6} + 296$$6$$1$$5$$C_6$$[\ ]_{6}$
37.6.5.5$x^{6} + 296$$6$$1$$5$$C_6$$[\ ]_{6}$
37.6.5.5$x^{6} + 296$$6$$1$$5$$C_6$$[\ ]_{6}$