↑

Properties

Label 18.0.14724425678...5712.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 7^{15}\cdot 19^{9}\cdot 31^{15}$
Root discriminant $612.51$
Ramified primes $2, 7, 19, 31$
Class number $36482973696$ (GRH)
Class group $[2, 2, 2, 4, 4, 4, 4, 12, 156, 9516]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8660565481, 38197162944, 75780975570, 90626337302, 74302857419, 45104419528, 21278039105, 8020398008, 2449633407, 608611962, 121957908, 19230652, 2249906, 162946, 1071, -1370, -127, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 127*x^16 - 1370*x^15 + 1071*x^14 + 162946*x^13 + 2249906*x^12 + 19230652*x^11 + 121957908*x^10 + 608611962*x^9 + 2449633407*x^8 + 8020398008*x^7 + 21278039105*x^6 + 45104419528*x^5 + 74302857419*x^4 + 90626337302*x^3 + 75780975570*x^2 + 38197162944*x + 8660565481)
 
gp: K = bnfinit(x^18 - 2*x^17 - 127*x^16 - 1370*x^15 + 1071*x^14 + 162946*x^13 + 2249906*x^12 + 19230652*x^11 + 121957908*x^10 + 608611962*x^9 + 2449633407*x^8 + 8020398008*x^7 + 21278039105*x^6 + 45104419528*x^5 + 74302857419*x^4 + 90626337302*x^3 + 75780975570*x^2 + 38197162944*x + 8660565481, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 127 x^{16} - 1370 x^{15} + 1071 x^{14} + 162946 x^{13} + 2249906 x^{12} + 19230652 x^{11} + 121957908 x^{10} + 608611962 x^{9} + 2449633407 x^{8} + 8020398008 x^{7} + 21278039105 x^{6} + 45104419528 x^{5} + 74302857419 x^{4} + 90626337302 x^{3} + 75780975570 x^{2} + 38197162944 x + 8660565481 \)

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:  \(-147244256784536162442327315331004890491666009075712=-\,2^{12}\cdot 7^{15}\cdot 19^{9}\cdot 31^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $612.51$
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:  $2, 7, 19, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{38} a^{12} - \frac{4}{19} a^{11} + \frac{7}{38} a^{10} - \frac{4}{19} a^{9} + \frac{1}{38} a^{8} + \frac{15}{38} a^{7} + \frac{2}{19} a^{6} - \frac{15}{38} a^{5} + \frac{6}{19} a^{4} + \frac{4}{19} a^{3} + \frac{9}{38} a^{2} + \frac{1}{38} a + \frac{8}{19}$, $\frac{1}{38} a^{13} - \frac{9}{38} a^{10} - \frac{3}{19} a^{9} - \frac{15}{38} a^{8} + \frac{5}{19} a^{7} - \frac{1}{19} a^{6} - \frac{13}{38} a^{5} + \frac{9}{38} a^{4} - \frac{3}{38} a^{3} + \frac{8}{19} a^{2} - \frac{7}{19} a + \frac{7}{19}$, $\frac{1}{1102} a^{14} + \frac{5}{1102} a^{13} + \frac{1}{551} a^{12} + \frac{111}{551} a^{11} + \frac{153}{1102} a^{10} - \frac{175}{1102} a^{9} - \frac{231}{551} a^{8} + \frac{267}{551} a^{7} - \frac{55}{551} a^{6} + \frac{71}{551} a^{5} - \frac{157}{551} a^{4} + \frac{75}{551} a^{3} - \frac{11}{1102} a^{2} - \frac{111}{1102} a + \frac{7}{19}$, $\frac{1}{239134} a^{15} - \frac{4}{119567} a^{14} + \frac{49}{34162} a^{13} - \frac{1236}{119567} a^{12} - \frac{5685}{34162} a^{11} - \frac{11145}{119567} a^{10} + \frac{811}{17081} a^{9} - \frac{72195}{239134} a^{8} + \frac{7473}{34162} a^{7} - \frac{57852}{119567} a^{6} - \frac{3655}{17081} a^{5} - \frac{1357}{12586} a^{4} - \frac{2342}{17081} a^{3} + \frac{7311}{239134} a^{2} + \frac{59617}{239134} a + \frac{3501}{8246}$, $\frac{1}{602378546} a^{16} - \frac{3}{301189273} a^{15} + \frac{41502}{301189273} a^{14} + \frac{2317101}{301189273} a^{13} - \frac{333005}{301189273} a^{12} - \frac{101119503}{602378546} a^{11} + \frac{39853014}{301189273} a^{10} + \frac{67473535}{602378546} a^{9} + \frac{13804601}{602378546} a^{8} + \frac{81978245}{602378546} a^{7} - \frac{153807691}{602378546} a^{6} + \frac{2251902}{9715783} a^{5} - \frac{142519820}{301189273} a^{4} - \frac{90473105}{301189273} a^{3} - \frac{845528}{301189273} a^{2} + \frac{180079043}{602378546} a - \frac{681134}{10385837}$, $\frac{1}{1622891165355165982493730133223181693460654} a^{17} - \frac{13343299599031842040971500509788}{811445582677582991246865066611590846730327} a^{16} - \frac{420849616818775005771964280358409357}{1622891165355165982493730133223181693460654} a^{15} + \frac{50140955151082934434364203712322784686}{115920797525368998749552152373084406675761} a^{14} - \frac{3613098903697439632544644820973099117595}{1622891165355165982493730133223181693460654} a^{13} + \frac{4314359304355088848015540120633388311987}{1622891165355165982493730133223181693460654} a^{12} - \frac{2606935915012364556217900953274641447553}{26175663957341386814415002148760995055817} a^{11} + \frac{262990181777724523985172490084133474673811}{1622891165355165982493730133223181693460654} a^{10} - \frac{26509267076542191557450615097486689767003}{1622891165355165982493730133223181693460654} a^{9} + \frac{177155328402339378774515668841301069233869}{1622891165355165982493730133223181693460654} a^{8} + \frac{36426155651478124478549722124788926218620}{811445582677582991246865066611590846730327} a^{7} - \frac{308095905503047852226848063464665746997145}{1622891165355165982493730133223181693460654} a^{6} - \frac{78252117438927758303413653192050554742678}{811445582677582991246865066611590846730327} a^{5} - \frac{647546343565163831559697495950074022522573}{1622891165355165982493730133223181693460654} a^{4} + \frac{6123245008957770250945370246734189326583}{55961764322591930430818280455971782533126} a^{3} + \frac{146332162270305749834867669433522373589603}{1622891165355165982493730133223181693460654} a^{2} - \frac{234336188608242602847954758479723520912557}{811445582677582991246865066611590846730327} a - \frac{445399310606828728008156799945555846813}{55961764322591930430818280455971782533126}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{4}\times C_{4}\times C_{12}\times C_{156}\times C_{9516}$, which has order $36482973696$ (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:  \( 36950112.38519628 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-4123}) \), 3.1.16492.1 x3, 3.3.47089.1, 6.0.1121398541872.1, Deg 6 x2, Deg 6, 9.3.9946239527177597280448.1 x3

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

Sibling fields

Degree 6 sibling: data not computed
Degree 9 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{18}$ R ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
2Data not computed
$7$7.6.5.6$x^{6} + 224$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.6$x^{6} + 224$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.6$x^{6} + 224$$6$$1$$5$$C_6$$[\ ]_{6}$
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
$31$31.6.5.4$x^{6} + 217$$6$$1$$5$$C_6$$[\ ]_{6}$
31.6.5.4$x^{6} + 217$$6$$1$$5$$C_6$$[\ ]_{6}$
31.6.5.4$x^{6} + 217$$6$$1$$5$$C_6$$[\ ]_{6}$