Properties

Label 18.0.11159062772...1168.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 7^{12}$
Root discriminant $10.06$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
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![1, -4, 14, -31, 64, -100, 146, -176, 202, -205, 202, -176, 146, -100, 64, -31, 14, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 14*x^16 - 31*x^15 + 64*x^14 - 100*x^13 + 146*x^12 - 176*x^11 + 202*x^10 - 205*x^9 + 202*x^8 - 176*x^7 + 146*x^6 - 100*x^5 + 64*x^4 - 31*x^3 + 14*x^2 - 4*x + 1)
gp: K = bnfinit(x^18 - 4*x^17 + 14*x^16 - 31*x^15 + 64*x^14 - 100*x^13 + 146*x^12 - 176*x^11 + 202*x^10 - 205*x^9 + 202*x^8 - 176*x^7 + 146*x^6 - 100*x^5 + 64*x^4 - 31*x^3 + 14*x^2 - 4*x + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut 4 x^{17} \) \(\mathstrut +\mathstrut 14 x^{16} \) \(\mathstrut -\mathstrut 31 x^{15} \) \(\mathstrut +\mathstrut 64 x^{14} \) \(\mathstrut -\mathstrut 100 x^{13} \) \(\mathstrut +\mathstrut 146 x^{12} \) \(\mathstrut -\mathstrut 176 x^{11} \) \(\mathstrut +\mathstrut 202 x^{10} \) \(\mathstrut -\mathstrut 205 x^{9} \) \(\mathstrut +\mathstrut 202 x^{8} \) \(\mathstrut -\mathstrut 176 x^{7} \) \(\mathstrut +\mathstrut 146 x^{6} \) \(\mathstrut -\mathstrut 100 x^{5} \) \(\mathstrut +\mathstrut 64 x^{4} \) \(\mathstrut -\mathstrut 31 x^{3} \) \(\mathstrut +\mathstrut 14 x^{2} \) \(\mathstrut -\mathstrut 4 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(-1115906277282951168=-\,2^{12}\cdot 3^{9}\cdot 7^{12}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $10.06$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{13} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{15} + \frac{4}{9} a^{13} + \frac{1}{3} a^{12} + \frac{1}{9} a^{11} + \frac{4}{9} a^{10} - \frac{4}{9} a^{9} - \frac{2}{9} a^{8} - \frac{4}{9} a^{7} + \frac{4}{9} a^{6} + \frac{1}{9} a^{5} + \frac{1}{3} a^{4} + \frac{4}{9} a^{3} + \frac{1}{9} a + \frac{1}{9}$, $\frac{1}{369} a^{17} - \frac{5}{369} a^{16} + \frac{20}{123} a^{15} - \frac{50}{369} a^{14} - \frac{44}{123} a^{13} - \frac{173}{369} a^{12} - \frac{50}{369} a^{11} + \frac{161}{369} a^{10} + \frac{2}{9} a^{9} - \frac{2}{9} a^{8} + \frac{79}{369} a^{7} - \frac{50}{369} a^{6} - \frac{44}{123} a^{5} + \frac{73}{369} a^{4} - \frac{1}{41} a^{3} + \frac{19}{369} a^{2} - \frac{128}{369} a + \frac{55}{123}$

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

Class group and class number

Trivial Abelian group, order $1$

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:  \( \frac{796}{369} a^{17} - \frac{616}{123} a^{16} + \frac{6104}{369} a^{15} - \frac{8066}{369} a^{14} + \frac{16616}{369} a^{13} - \frac{12617}{369} a^{12} + \frac{2074}{41} a^{11} - \frac{7226}{369} a^{10} + \frac{256}{9} a^{9} - \frac{4}{9} a^{8} + \frac{1466}{369} a^{7} + \frac{3598}{123} a^{6} - \frac{6754}{369} a^{5} + \frac{14566}{369} a^{4} - \frac{5770}{369} a^{3} + \frac{7006}{369} a^{2} - \frac{616}{123} a + \frac{1370}{369} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{572}{369} a^{17} - \frac{1138}{369} a^{16} + \frac{1231}{123} a^{15} - \frac{3877}{369} a^{14} + \frac{2794}{123} a^{13} - \frac{3508}{369} a^{12} + \frac{6578}{369} a^{11} + \frac{457}{369} a^{10} + \frac{64}{9} a^{9} + \frac{32}{9} a^{8} + \frac{170}{369} a^{7} + \frac{6332}{369} a^{6} - \frac{650}{123} a^{5} + \frac{5102}{369} a^{4} + \frac{47}{123} a^{3} + \frac{167}{369} a^{2} + \frac{215}{369} a - \frac{23}{41} \),  \( \frac{737}{369} a^{17} - \frac{914}{123} a^{16} + \frac{8878}{369} a^{15} - \frac{17908}{369} a^{14} + \frac{34285}{369} a^{13} - \frac{48658}{369} a^{12} + \frac{21610}{123} a^{11} - \frac{71788}{369} a^{10} + \frac{1880}{9} a^{9} - \frac{1838}{9} a^{8} + \frac{70810}{369} a^{7} - \frac{6313}{41} a^{6} + \frac{42157}{369} a^{5} - \frac{24919}{369} a^{4} + \frac{14113}{369} a^{3} - \frac{6415}{369} a^{2} + \frac{685}{123} a - \frac{575}{369} \),  \( a \),  \( \frac{499}{369} a^{17} - \frac{367}{123} a^{16} + \frac{3659}{369} a^{15} - \frac{4163}{369} a^{14} + \frac{8342}{369} a^{13} - \frac{2564}{369} a^{12} + \frac{362}{41} a^{11} + \frac{9778}{369} a^{10} - \frac{245}{9} a^{9} + \frac{479}{9} a^{8} - \frac{17446}{369} a^{7} + \frac{8261}{123} a^{6} - \frac{19825}{369} a^{5} + \frac{22774}{369} a^{4} - \frac{11584}{369} a^{3} + \frac{8620}{369} a^{2} - \frac{218}{41} a + \frac{827}{369} \),  \( \frac{43}{369} a^{17} + \frac{605}{369} a^{16} - \frac{2381}{369} a^{15} + \frac{8428}{369} a^{14} - \frac{17279}{369} a^{13} + \frac{34135}{369} a^{12} - \frac{48562}{369} a^{11} + \frac{21769}{123} a^{10} - \frac{575}{3} a^{9} + \frac{613}{3} a^{8} - \frac{23741}{123} a^{7} + \frac{67673}{369} a^{6} - \frac{52580}{369} a^{5} + \frac{38809}{369} a^{4} - \frac{20354}{369} a^{3} + \frac{10903}{369} a^{2} - \frac{3454}{369} a + \frac{1519}{369} \),  \( \frac{62}{123} a^{17} - \frac{1094}{369} a^{16} + \frac{3862}{369} a^{15} - \frac{1088}{41} a^{14} + \frac{19933}{369} a^{13} - \frac{11300}{123} a^{12} + \frac{48715}{369} a^{11} - \frac{60992}{369} a^{10} + \frac{1678}{9} a^{9} - \frac{1720}{9} a^{8} + \frac{68609}{369} a^{7} - \frac{60755}{369} a^{6} + \frac{48838}{369} a^{5} - \frac{11300}{123} a^{4} + \frac{20056}{369} a^{3} - \frac{2963}{123} a^{2} + \frac{3580}{369} a - \frac{593}{369} \),  \( \frac{392}{369} a^{17} - \frac{2206}{369} a^{16} + \frac{2387}{123} a^{15} - \frac{17263}{369} a^{14} + \frac{10796}{123} a^{13} - \frac{52564}{369} a^{12} + \frac{68468}{369} a^{11} - \frac{81905}{369} a^{10} + \frac{2065}{9} a^{9} - \frac{2080}{9} a^{8} + \frac{78692}{369} a^{7} - \frac{67570}{369} a^{6} + \frac{15962}{123} a^{5} - \frac{30670}{369} a^{4} + \frac{4810}{123} a^{3} - \frac{6328}{369} a^{2} + \frac{1730}{369} a - \frac{2}{41} \),  \( \frac{509}{123} a^{17} - \frac{5462}{369} a^{16} + \frac{17656}{369} a^{15} - \frac{11633}{123} a^{14} + \frac{66658}{369} a^{13} - \frac{10260}{41} a^{12} + \frac{121393}{369} a^{11} - \frac{130820}{369} a^{10} + \frac{3331}{9} a^{9} - \frac{3163}{9} a^{8} + \frac{119444}{369} a^{7} - \frac{91766}{369} a^{6} + \frac{65674}{369} a^{5} - \frac{3741}{41} a^{4} + \frac{15613}{369} a^{3} - \frac{603}{41} a^{2} + \frac{1057}{369} a - \frac{359}{369} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 64.801283476 \)
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

\(\Q(\sqrt{-3}) \), 3.1.588.1 x3, \(\Q(\zeta_{7})^+\), 6.0.1037232.1, 6.0.21168.1 x2, 6.0.64827.1, 9.3.203297472.1 x3

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

Sibling fields

Degree 6 sibling: 6.0.21168.1
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 R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$