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Properties

Label 18.0.43028548344...9168.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 7^{12}\cdot 17^{9}$
Root discriminant $30.18$
Ramified primes $2, 7, 17$
Class number $12$
Class group $[12]$
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![46984, 37800, 82964, 85532, 54332, 18420, 12797, -4982, -4843, -2970, 1159, -154, -85, -26, 95, 2, -11, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 11*x^16 + 2*x^15 + 95*x^14 - 26*x^13 - 85*x^12 - 154*x^11 + 1159*x^10 - 2970*x^9 - 4843*x^8 - 4982*x^7 + 12797*x^6 + 18420*x^5 + 54332*x^4 + 85532*x^3 + 82964*x^2 + 37800*x + 46984)
 
gp: K = bnfinit(x^18 - 2*x^17 - 11*x^16 + 2*x^15 + 95*x^14 - 26*x^13 - 85*x^12 - 154*x^11 + 1159*x^10 - 2970*x^9 - 4843*x^8 - 4982*x^7 + 12797*x^6 + 18420*x^5 + 54332*x^4 + 85532*x^3 + 82964*x^2 + 37800*x + 46984, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 11 x^{16} + 2 x^{15} + 95 x^{14} - 26 x^{13} - 85 x^{12} - 154 x^{11} + 1159 x^{10} - 2970 x^{9} - 4843 x^{8} - 4982 x^{7} + 12797 x^{6} + 18420 x^{5} + 54332 x^{4} + 85532 x^{3} + 82964 x^{2} + 37800 x + 46984 \)

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:  \(-430285483449173885556359168=-\,2^{18}\cdot 7^{12}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.18$
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:  $2, 7, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{7} a^{11} - \frac{3}{7} a^{10} + \frac{3}{7} a^{9} - \frac{1}{7} a^{8} + \frac{3}{7} a^{7} + \frac{3}{7} a^{6} - \frac{1}{7} a^{5} - \frac{3}{7} a^{4} - \frac{3}{7} a^{3}$, $\frac{1}{21} a^{12} - \frac{2}{7} a^{10} - \frac{2}{7} a^{9} + \frac{1}{3} a^{8} - \frac{3}{7} a^{7} - \frac{2}{7} a^{6} - \frac{2}{7} a^{5} + \frac{3}{7} a^{4} - \frac{3}{7} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{21} a^{13} - \frac{1}{7} a^{10} + \frac{4}{21} a^{9} + \frac{2}{7} a^{8} - \frac{3}{7} a^{7} - \frac{3}{7} a^{6} + \frac{1}{7} a^{5} - \frac{2}{7} a^{4} + \frac{10}{21} a^{3} + \frac{1}{3} a$, $\frac{1}{42} a^{14} - \frac{1}{42} a^{12} + \frac{1}{42} a^{10} - \frac{19}{42} a^{8} - \frac{2}{7} a^{7} - \frac{1}{14} a^{6} + \frac{3}{7} a^{5} + \frac{13}{42} a^{4} - \frac{1}{2} a^{2} + \frac{1}{3}$, $\frac{1}{882} a^{15} - \frac{1}{126} a^{14} - \frac{1}{294} a^{13} - \frac{5}{882} a^{12} - \frac{23}{882} a^{11} + \frac{101}{882} a^{10} + \frac{89}{294} a^{9} + \frac{1}{126} a^{8} - \frac{13}{98} a^{7} - \frac{65}{294} a^{6} - \frac{233}{882} a^{5} + \frac{53}{882} a^{4} + \frac{433}{882} a^{3} + \frac{5}{42} a^{2} - \frac{3}{7} a - \frac{25}{63}$, $\frac{1}{12348} a^{16} + \frac{1}{3087} a^{15} + \frac{67}{12348} a^{14} + \frac{22}{3087} a^{13} - \frac{89}{4116} a^{12} + \frac{151}{3087} a^{11} - \frac{5279}{12348} a^{10} - \frac{272}{3087} a^{9} + \frac{5693}{12348} a^{8} + \frac{76}{1029} a^{7} + \frac{5623}{12348} a^{6} - \frac{659}{3087} a^{5} - \frac{1357}{12348} a^{4} + \frac{355}{6174} a^{3} + \frac{127}{294} a^{2} - \frac{5}{63} a + \frac{20}{441}$, $\frac{1}{353598958145122294720656330245988} a^{17} + \frac{95613917521973209222850186}{12628534219468653382880583223071} a^{16} + \frac{191396339948581010772588161909}{353598958145122294720656330245988} a^{15} - \frac{392111841458301705610858667405}{176799479072561147360328165122994} a^{14} - \frac{7853503250716703483030824189097}{353598958145122294720656330245988} a^{13} + \frac{3989770298389042827871380738845}{176799479072561147360328165122994} a^{12} - \frac{246907320529490796712750364209}{353598958145122294720656330245988} a^{11} - \frac{1598393176764987956814536031491}{176799479072561147360328165122994} a^{10} - \frac{6486222757741135902977098324817}{16838045625958204510507444297428} a^{9} - \frac{49601297451707169693840988224023}{176799479072561147360328165122994} a^{8} - \frac{10111469326496111131503623593019}{353598958145122294720656330245988} a^{7} - \frac{18134512014547963760734445563769}{176799479072561147360328165122994} a^{6} - \frac{25735861222791608802570839669969}{117866319381707431573552110081996} a^{5} - \frac{13204767750290047808057001515732}{88399739536280573680164082561497} a^{4} - \frac{37653634090144965395012266498612}{88399739536280573680164082561497} a^{3} + \frac{3255799136188678209419056765475}{25257068438937306765761166446142} a^{2} + \frac{438985030713515450510003198908}{1403170468829850375875620358119} a + \frac{2821538033103650735269933642556}{12628534219468653382880583223071}$

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

Class group and class number

$C_{12}$, which has order $12$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 117049.901978 \)
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

\(\Q(\sqrt{-17}) \), 3.1.3332.1 x3, \(\Q(\zeta_{7})^+\), 6.0.754951232.1, 6.0.754951232.2, 6.0.15407168.1 x2, 9.3.36992610368.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.15407168.1
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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
$17$17.6.3.1$x^{6} - 34 x^{4} + 289 x^{2} - 44217$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.1$x^{6} - 34 x^{4} + 289 x^{2} - 44217$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.1$x^{6} - 34 x^{4} + 289 x^{2} - 44217$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.2e2_17.2t1.1c1$1$ $ 2^{2} \cdot 17 $ $x^{2} + 17$ $C_2$ (as 2T1) $1$ $-1$
* 1.2e2_7_17.6t1.2c1$1$ $ 2^{2} \cdot 7 \cdot 17 $ $x^{6} - 2 x^{5} + 48 x^{4} - 62 x^{3} + 903 x^{2} - 616 x + 6461$ $C_6$ (as 6T1) $0$ $-1$
* 1.2e2_7_17.6t1.2c2$1$ $ 2^{2} \cdot 7 \cdot 17 $ $x^{6} - 2 x^{5} + 48 x^{4} - 62 x^{3} + 903 x^{2} - 616 x + 6461$ $C_6$ (as 6T1) $0$ $-1$
* 1.7.3t1.1c1$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
* 1.7.3t1.1c2$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
*2 2.2e2_7e2_17.3t2.1c1$2$ $ 2^{2} \cdot 7^{2} \cdot 17 $ $x^{3} - x^{2} + 12 x - 20$ $S_3$ (as 3T2) $1$ $0$
*2 2.2e2_7_17.6t5.3c1$2$ $ 2^{2} \cdot 7 \cdot 17 $ $x^{18} - 2 x^{17} - 11 x^{16} + 2 x^{15} + 95 x^{14} - 26 x^{13} - 85 x^{12} - 154 x^{11} + 1159 x^{10} - 2970 x^{9} - 4843 x^{8} - 4982 x^{7} + 12797 x^{6} + 18420 x^{5} + 54332 x^{4} + 85532 x^{3} + 82964 x^{2} + 37800 x + 46984$ $S_3 \times C_3$ (as 18T3) $0$ $0$
*2 2.2e2_7_17.6t5.3c2$2$ $ 2^{2} \cdot 7 \cdot 17 $ $x^{18} - 2 x^{17} - 11 x^{16} + 2 x^{15} + 95 x^{14} - 26 x^{13} - 85 x^{12} - 154 x^{11} + 1159 x^{10} - 2970 x^{9} - 4843 x^{8} - 4982 x^{7} + 12797 x^{6} + 18420 x^{5} + 54332 x^{4} + 85532 x^{3} + 82964 x^{2} + 37800 x + 46984$ $S_3 \times C_3$ (as 18T3) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.