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Properties

Label 18.0.53072791277...0608.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 37^{12}$
Root discriminant $30.53$
Ramified primes $2, 3, 37$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group $S_3^2$ (as 18T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, -32, 112, -112, 200, -144, 216, -112, 130, -126, -17, 17, 63, -28, -12, 6, 5, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 5*x^16 + 6*x^15 - 12*x^14 - 28*x^13 + 63*x^12 + 17*x^11 - 17*x^10 - 126*x^9 + 130*x^8 - 112*x^7 + 216*x^6 - 144*x^5 + 200*x^4 - 112*x^3 + 112*x^2 - 32*x + 64)
 
gp: K = bnfinit(x^18 - 3*x^17 + 5*x^16 + 6*x^15 - 12*x^14 - 28*x^13 + 63*x^12 + 17*x^11 - 17*x^10 - 126*x^9 + 130*x^8 - 112*x^7 + 216*x^6 - 144*x^5 + 200*x^4 - 112*x^3 + 112*x^2 - 32*x + 64, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 5 x^{16} + 6 x^{15} - 12 x^{14} - 28 x^{13} + 63 x^{12} + 17 x^{11} - 17 x^{10} - 126 x^{9} + 130 x^{8} - 112 x^{7} + 216 x^{6} - 144 x^{5} + 200 x^{4} - 112 x^{3} + 112 x^{2} - 32 x + 64 \)

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:  \(-530727912779568801529540608=-\,2^{12}\cdot 3^{9}\cdot 37^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.53$
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:  $2, 3, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{11} - \frac{1}{12} a^{10} - \frac{1}{6} a^{9} + \frac{1}{6} a^{8} + \frac{1}{6} a^{7} + \frac{5}{12} a^{6} + \frac{1}{12} a^{5} - \frac{1}{12} a^{4} - \frac{1}{3} a^{3} + \frac{1}{6} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{12} a^{13} + \frac{1}{12} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{12} a^{8} - \frac{1}{6} a^{7} + \frac{1}{4} a^{6} - \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{24} a^{14} - \frac{1}{24} a^{13} - \frac{1}{24} a^{12} - \frac{1}{12} a^{11} - \frac{1}{6} a^{10} + \frac{1}{12} a^{9} + \frac{5}{24} a^{8} + \frac{1}{24} a^{7} - \frac{1}{24} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a^{3} + \frac{1}{6} a^{2}$, $\frac{1}{528} a^{15} + \frac{3}{176} a^{14} - \frac{1}{528} a^{13} - \frac{1}{66} a^{12} - \frac{3}{88} a^{11} + \frac{7}{44} a^{10} + \frac{35}{528} a^{9} - \frac{9}{176} a^{8} + \frac{7}{176} a^{7} - \frac{17}{44} a^{6} - \frac{4}{33} a^{5} + \frac{3}{11} a^{4} + \frac{13}{66} a^{3} + \frac{1}{22} a^{2} + \frac{9}{22} a - \frac{4}{33}$, $\frac{1}{40128} a^{16} - \frac{37}{40128} a^{15} + \frac{23}{13376} a^{14} - \frac{43}{1056} a^{13} - \frac{15}{6688} a^{12} - \frac{553}{10032} a^{11} - \frac{1189}{40128} a^{10} - \frac{3661}{40128} a^{9} - \frac{1383}{13376} a^{8} - \frac{4457}{20064} a^{7} + \frac{1297}{5016} a^{6} + \frac{2741}{10032} a^{5} + \frac{92}{627} a^{4} - \frac{653}{1672} a^{3} + \frac{439}{5016} a^{2} + \frac{13}{2508} a + \frac{137}{418}$, $\frac{1}{2017996992} a^{17} - \frac{257}{45863568} a^{16} - \frac{9929}{21020802} a^{15} - \frac{9900479}{672665664} a^{14} - \frac{2244273}{56055472} a^{13} + \frac{30254617}{1008998496} a^{12} - \frac{14144681}{2017996992} a^{11} + \frac{132197579}{1008998496} a^{10} - \frac{728899}{336332832} a^{9} - \frac{35611199}{224221888} a^{8} + \frac{96953663}{1008998496} a^{7} - \frac{44698865}{504499248} a^{6} + \frac{194719361}{504499248} a^{5} - \frac{1014715}{252249624} a^{4} - \frac{24684427}{63062406} a^{3} + \frac{44865725}{252249624} a^{2} + \frac{18713857}{42041604} a + \frac{16424939}{63062406}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  \( -\frac{436853}{106210368} a^{17} + \frac{223783}{53105184} a^{16} + \frac{70715}{17701728} a^{15} - \frac{2309771}{35403456} a^{14} + \frac{5549}{8850864} a^{13} + \frac{11370493}{53105184} a^{12} - \frac{2301059}{106210368} a^{11} - \frac{7858363}{13276296} a^{10} - \frac{789121}{8850864} a^{9} + \frac{8697365}{11801152} a^{8} + \frac{29363429}{53105184} a^{7} - \frac{21637529}{26552592} a^{6} + \frac{178121}{26552592} a^{5} - \frac{8599177}{13276296} a^{4} + \frac{2852065}{6638148} a^{3} - \frac{1799105}{1206936} a^{2} + \frac{209917}{737572} a + \frac{26831}{3319074} \) (order $6$)
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:  \( 3307615.5705449595 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.5476.1, 3.1.4107.1 x3, 6.0.809637552.2, 6.0.50602347.2, 9.1.4433575234752.1 x3

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 6 sibling: data not computed
Degree 9 sibling: data not computed
Degree 12 sibling: data not computed
Degree 18 siblings: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$37$37.9.6.1$x^{9} + 222 x^{6} + 15059 x^{3} + 405224$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
37.9.6.1$x^{9} + 222 x^{6} + 15059 x^{3} + 405224$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$