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Properties

Label 18.0.22300105675...6176.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{31}\cdot 7^{14}\cdot 37^{9}$
Root discriminant $290.93$
Ramified primes $2, 3, 7, 37$
Class number $296522208$ (GRH)
Class group $[3, 6, 6, 2745576]$ (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![266589164359, -137278953357, 116703097728, -4634433048, 11832099909, -1511623617, 3126531786, -161530719, 437841852, -10748977, 29917974, -574347, 1092552, -26121, 20739, -372, 210, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 210*x^16 - 372*x^15 + 20739*x^14 - 26121*x^13 + 1092552*x^12 - 574347*x^11 + 29917974*x^10 - 10748977*x^9 + 437841852*x^8 - 161530719*x^7 + 3126531786*x^6 - 1511623617*x^5 + 11832099909*x^4 - 4634433048*x^3 + 116703097728*x^2 - 137278953357*x + 266589164359)
 
gp: K = bnfinit(x^18 - 3*x^17 + 210*x^16 - 372*x^15 + 20739*x^14 - 26121*x^13 + 1092552*x^12 - 574347*x^11 + 29917974*x^10 - 10748977*x^9 + 437841852*x^8 - 161530719*x^7 + 3126531786*x^6 - 1511623617*x^5 + 11832099909*x^4 - 4634433048*x^3 + 116703097728*x^2 - 137278953357*x + 266589164359, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 210 x^{16} - 372 x^{15} + 20739 x^{14} - 26121 x^{13} + 1092552 x^{12} - 574347 x^{11} + 29917974 x^{10} - 10748977 x^{9} + 437841852 x^{8} - 161530719 x^{7} + 3126531786 x^{6} - 1511623617 x^{5} + 11832099909 x^{4} - 4634433048 x^{3} + 116703097728 x^{2} - 137278953357 x + 266589164359 \)

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:  \(-223001056753368049126899632782755242066866176=-\,2^{12}\cdot 3^{31}\cdot 7^{14}\cdot 37^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $290.93$
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:  $2, 3, 7, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{2} - \frac{1}{9} a + \frac{1}{9}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{9} + \frac{1}{9} a^{3} + \frac{1}{9}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{10} + \frac{1}{9} a^{4} + \frac{1}{9} a$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{1}{9} a^{5} + \frac{1}{9} a - \frac{1}{9}$, $\frac{1}{63} a^{15} - \frac{1}{21} a^{14} - \frac{1}{63} a^{12} - \frac{2}{63} a^{11} - \frac{4}{63} a^{10} - \frac{1}{63} a^{9} + \frac{1}{21} a^{8} + \frac{1}{7} a^{7} - \frac{8}{63} a^{6} + \frac{1}{3} a^{5} - \frac{4}{9} a^{3} + \frac{4}{9} a^{2} + \frac{2}{9} a - \frac{4}{9}$, $\frac{1}{189} a^{16} + \frac{1}{189} a^{15} + \frac{1}{21} a^{14} - \frac{1}{189} a^{13} + \frac{8}{189} a^{12} + \frac{1}{21} a^{11} - \frac{17}{189} a^{10} - \frac{8}{189} a^{9} + \frac{4}{27} a^{7} + \frac{10}{189} a^{6} + \frac{1}{3} a^{5} - \frac{4}{27} a^{4} - \frac{4}{27} a^{3} + \frac{1}{3} a^{2} + \frac{13}{27} a + \frac{4}{27}$, $\frac{1}{50093935124108768345570024038590788940515649077925101113197750019607825774847519739278597} a^{17} + \frac{6148217792436004451497083191499345393499915479911635302966291999128207460394168664000}{50093935124108768345570024038590788940515649077925101113197750019607825774847519739278597} a^{16} - \frac{31718382705419534093736557288072824989416387672575517475778541714127710776668463342087}{4553994102191706213233638548962798994592331734356827373927068183600711434077047249025327} a^{15} + \frac{807737539034160621086575467286727952278226886871940035152559070165508768925363359042718}{50093935124108768345570024038590788940515649077925101113197750019607825774847519739278597} a^{14} + \frac{635194918555202091506488945275898394045340853705526861987868272652821265930037877641837}{50093935124108768345570024038590788940515649077925101113197750019607825774847519739278597} a^{13} + \frac{2476558175323675924495076318579548386987088795466475087096623018513507662406746900523276}{50093935124108768345570024038590788940515649077925101113197750019607825774847519739278597} a^{12} + \frac{72696934970389798132284185903116659077850017447072383473073085897938857685322217425251}{50093935124108768345570024038590788940515649077925101113197750019607825774847519739278597} a^{11} - \frac{1200996627387177787489026365666424988029421447994100670334827928645444915686006517141719}{50093935124108768345570024038590788940515649077925101113197750019607825774847519739278597} a^{10} - \frac{6044959180909053413036416697784011575710967698834951928147822189593984000862892783389880}{50093935124108768345570024038590788940515649077925101113197750019607825774847519739278597} a^{9} - \frac{6794791549047533791031966811571777248870260298906614415312043203852187333874463535979762}{50093935124108768345570024038590788940515649077925101113197750019607825774847519739278597} a^{8} - \frac{658754859491677847544151455752393525135970019304322354195445561209286891351036100436950}{4553994102191706213233638548962798994592331734356827373927068183600711434077047249025327} a^{7} + \frac{5989806383813486001498105223955134032853438116628922582994925925908463158251576586038205}{50093935124108768345570024038590788940515649077925101113197750019607825774847519739278597} a^{6} - \frac{720227702724829603889256859297019845682418210965893871798473826624279644770353540985796}{7156276446301252620795717719798684134359378439703585873313964288515403682121074248468371} a^{5} - \frac{3217687940899692161811656723157977566567721135873318533539745172556161184018732060409217}{7156276446301252620795717719798684134359378439703585873313964288515403682121074248468371} a^{4} - \frac{246100065909368141297112675386770763488896855000253906161598953421324847279009817057686}{650570586027386601890519792708971284941761676336689624846724026228673062011006749860761} a^{3} - \frac{2261825121781246261440977234410441784257045779908759278215306886348824718133787122207183}{7156276446301252620795717719798684134359378439703585873313964288515403682121074248468371} a^{2} - \frac{2402757422604146126453489961907818752570656638411992650231388759410891579059297711894640}{7156276446301252620795717719798684134359378439703585873313964288515403682121074248468371} a + \frac{3273831596849062630292884486262909549565863588725521332730911808925195594152214822879510}{7156276446301252620795717719798684134359378439703585873313964288515403682121074248468371}$

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

Class group and class number

$C_{3}\times C_{6}\times C_{6}\times C_{2745576}$, which has order $296522208$ (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:  \( 4695974.091249611 \) (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

\(\Q(\sqrt{-111}) \), 3.3.3969.2, 3.3.756.1, 6.0.86850039024.2, 6.0.2393804200599.8, 9.9.756284282720064.2

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 R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
3Data not computed
$7$7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.4$x^{6} + 14$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.4$x^{6} + 14$$6$$1$$5$$C_6$$[\ ]_{6}$
$37$37.6.3.1$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
37.6.3.1$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
37.6.3.1$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$