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Properties

Label 18.0.69006009923...6032.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 37^{14}\cdot 83^{9}$
Root discriminant $239.85$
Ramified primes $2, 37, 83$
Class number $153823860$ (GRH)
Class group $[3, 6, 8545770]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2368812673291, -799383089986, 845688187315, -221021307185, 129902924239, -27289998619, 11609272935, -1995570269, 679813833, -96044632, 27539182, -3170897, 783481, -71437, 15239, -1018, 183, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 183*x^16 - 1018*x^15 + 15239*x^14 - 71437*x^13 + 783481*x^12 - 3170897*x^11 + 27539182*x^10 - 96044632*x^9 + 679813833*x^8 - 1995570269*x^7 + 11609272935*x^6 - 27289998619*x^5 + 129902924239*x^4 - 221021307185*x^3 + 845688187315*x^2 - 799383089986*x + 2368812673291)
 
gp: K = bnfinit(x^18 - 7*x^17 + 183*x^16 - 1018*x^15 + 15239*x^14 - 71437*x^13 + 783481*x^12 - 3170897*x^11 + 27539182*x^10 - 96044632*x^9 + 679813833*x^8 - 1995570269*x^7 + 11609272935*x^6 - 27289998619*x^5 + 129902924239*x^4 - 221021307185*x^3 + 845688187315*x^2 - 799383089986*x + 2368812673291, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 183 x^{16} - 1018 x^{15} + 15239 x^{14} - 71437 x^{13} + 783481 x^{12} - 3170897 x^{11} + 27539182 x^{10} - 96044632 x^{9} + 679813833 x^{8} - 1995570269 x^{7} + 11609272935 x^{6} - 27289998619 x^{5} + 129902924239 x^{4} - 221021307185 x^{3} + 845688187315 x^{2} - 799383089986 x + 2368812673291 \)

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:  \(-6900600992399474037809909297898044762796032=-\,2^{12}\cdot 37^{14}\cdot 83^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $239.85$
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:  $2, 37, 83$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{72340301059730490658510892347424875104315163568769436421232794016333} a^{17} - \frac{156504904217578073621114812633743369390475270483568202562832750489}{72340301059730490658510892347424875104315163568769436421232794016333} a^{16} + \frac{1875535032111826502135654191973233719600672778252973414800747020850}{72340301059730490658510892347424875104315163568769436421232794016333} a^{15} + \frac{5780953742018040882287630505007152841918859460934029221036410775549}{72340301059730490658510892347424875104315163568769436421232794016333} a^{14} + \frac{12009308572539056679311882215684985112080037061130909579096477062160}{72340301059730490658510892347424875104315163568769436421232794016333} a^{13} + \frac{5780780766377535336790244375524428924463747318715446216596150323752}{72340301059730490658510892347424875104315163568769436421232794016333} a^{12} - \frac{19124742729520993569195795317884105135315173428230813848891183021170}{72340301059730490658510892347424875104315163568769436421232794016333} a^{11} + \frac{21666081290750831682919151667975952256897397346804320970609362970804}{72340301059730490658510892347424875104315163568769436421232794016333} a^{10} + \frac{5146511662814659632499273894186034939235488908300697459743465571800}{72340301059730490658510892347424875104315163568769436421232794016333} a^{9} + \frac{10161469575118709328480954517494448863691194862286346127250262327302}{72340301059730490658510892347424875104315163568769436421232794016333} a^{8} - \frac{29261949381542717722131483479810359182009185222699779755348269203333}{72340301059730490658510892347424875104315163568769436421232794016333} a^{7} + \frac{22038864948553330158426546304484283970402294128922063616797613305079}{72340301059730490658510892347424875104315163568769436421232794016333} a^{6} - \frac{7740662541748472993411846270575758504710863011030119861019005828048}{24113433686576830219503630782474958368105054522923145473744264672111} a^{5} + \frac{20466430530183829563040854238658607601860773421552000433740758027975}{72340301059730490658510892347424875104315163568769436421232794016333} a^{4} - \frac{16415132362557454033049033642385260505651947355544180919709514571516}{72340301059730490658510892347424875104315163568769436421232794016333} a^{3} - \frac{6059560288575672700480655170866890458065701724289725588125896249315}{24113433686576830219503630782474958368105054522923145473744264672111} a^{2} - \frac{31087859422698556298047485986762428224174039536043400841196442824430}{72340301059730490658510892347424875104315163568769436421232794016333} a - \frac{20392507428319202297239062372788834298937506962112129079186084112288}{72340301059730490658510892347424875104315163568769436421232794016333}$

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

Class group and class number

$C_{3}\times C_{6}\times C_{8545770}$, which has order $153823860$ (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:  \( 615797.1340659427 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-83}) \), 3.3.1369.1, 3.3.148.1, 6.0.1071620895707.2, 6.0.12524422448.5, 9.9.6075640136512.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
2Data not computed
$37$37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$
37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$
$83$83.6.3.2$x^{6} - 6889 x^{2} + 1715361$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
83.6.3.2$x^{6} - 6889 x^{2} + 1715361$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
83.6.3.2$x^{6} - 6889 x^{2} + 1715361$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$