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Properties

Label 18.0.95110687742...0016.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{44}\cdot 11^{9}$
Root discriminant $77.21$
Ramified primes $2, 3, 11$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2\times He_3:C_2$ (as 18T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![106415, -211203, 399015, -898146, 1053576, -700686, 436710, -310590, 196839, -108675, 53955, -24174, 9780, -3492, 1116, -288, 63, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 63*x^16 - 288*x^15 + 1116*x^14 - 3492*x^13 + 9780*x^12 - 24174*x^11 + 53955*x^10 - 108675*x^9 + 196839*x^8 - 310590*x^7 + 436710*x^6 - 700686*x^5 + 1053576*x^4 - 898146*x^3 + 399015*x^2 - 211203*x + 106415)
 
gp: K = bnfinit(x^18 - 9*x^17 + 63*x^16 - 288*x^15 + 1116*x^14 - 3492*x^13 + 9780*x^12 - 24174*x^11 + 53955*x^10 - 108675*x^9 + 196839*x^8 - 310590*x^7 + 436710*x^6 - 700686*x^5 + 1053576*x^4 - 898146*x^3 + 399015*x^2 - 211203*x + 106415, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 63 x^{16} - 288 x^{15} + 1116 x^{14} - 3492 x^{13} + 9780 x^{12} - 24174 x^{11} + 53955 x^{10} - 108675 x^{9} + 196839 x^{8} - 310590 x^{7} + 436710 x^{6} - 700686 x^{5} + 1053576 x^{4} - 898146 x^{3} + 399015 x^{2} - 211203 x + 106415 \)

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:  \(-9511068774268243823056556964950016=-\,2^{12}\cdot 3^{44}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $77.21$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{6} a^{10} + \frac{1}{6} a^{8} + \frac{1}{6} a^{6} - \frac{1}{6} a^{4} - \frac{1}{6} a^{2} - \frac{1}{6}$, $\frac{1}{6} a^{11} + \frac{1}{6} a^{9} + \frac{1}{6} a^{7} - \frac{1}{6} a^{5} - \frac{1}{6} a^{3} - \frac{1}{6} a$, $\frac{1}{12} a^{12} - \frac{1}{12} a^{10} + \frac{1}{6} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{5}{12} a^{2} - \frac{1}{2} a - \frac{1}{12}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{11} + \frac{1}{6} a^{9} - \frac{1}{4} a^{7} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} + \frac{1}{12} a^{3} - \frac{1}{2} a^{2} + \frac{5}{12} a - \frac{1}{2}$, $\frac{1}{24} a^{14} - \frac{1}{24} a^{13} + \frac{1}{24} a^{11} - \frac{1}{24} a^{10} + \frac{1}{6} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} - \frac{1}{24} a^{6} - \frac{1}{6} a^{5} - \frac{11}{24} a^{4} - \frac{7}{24} a^{3} + \frac{1}{12} a^{2} - \frac{11}{24} a + \frac{1}{24}$, $\frac{1}{24} a^{15} - \frac{1}{24} a^{13} - \frac{1}{24} a^{12} + \frac{1}{24} a^{10} + \frac{1}{24} a^{9} - \frac{1}{12} a^{8} - \frac{1}{6} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} + \frac{1}{12} a^{4} + \frac{7}{24} a^{3} + \frac{5}{24} a^{2} + \frac{1}{12} a - \frac{5}{24}$, $\frac{1}{48} a^{16} - \frac{1}{12} a^{10} - \frac{1}{8} a^{9} - \frac{11}{48} a^{8} - \frac{1}{4} a^{7} + \frac{1}{12} a^{6} + \frac{1}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{12} a^{2} - \frac{1}{8} a + \frac{5}{48}$, $\frac{1}{341233907673233872165503177616122320784} a^{17} + \frac{2964426628379334290772490957116944513}{341233907673233872165503177616122320784} a^{16} - \frac{3387847786463347537364127350597147}{7109039743192372336781316200335881683} a^{15} - \frac{993048173804259336300145838176701529}{85308476918308468041375794404030580196} a^{14} + \frac{264680296946565814723960014077569141}{7109039743192372336781316200335881683} a^{13} + \frac{823436479976121976814306935268914396}{21327119229577117010343948601007645049} a^{12} - \frac{3395815684471835301463894371129134449}{85308476918308468041375794404030580196} a^{11} + \frac{1642854423959889841730346916880933917}{170616953836616936082751588808061160392} a^{10} + \frac{46575447657686180871182701923003139279}{341233907673233872165503177616122320784} a^{9} + \frac{34485208673149419146903736257687339741}{341233907673233872165503177616122320784} a^{8} - \frac{3169882239504666926560010315314197191}{21327119229577117010343948601007645049} a^{7} - \frac{38050286804573263911001160846794484905}{170616953836616936082751588808061160392} a^{6} - \frac{26396432251412428766087029464152149497}{56872317945538978694250529602687053464} a^{5} - \frac{28583717535162266960120538367959293317}{85308476918308468041375794404030580196} a^{4} - \frac{5156140337245859301812390639567537537}{42654238459154234020687897202015290098} a^{3} + \frac{58025930040366139613188528015462923499}{170616953836616936082751588808061160392} a^{2} + \frac{58934975393031513328580274146169859631}{341233907673233872165503177616122320784} a - \frac{57048376989748661492788602574221746015}{341233907673233872165503177616122320784}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 7037853377.607728 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3:C_2$ (as 18T41):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times He_3:C_2$
Character table for $C_2\times He_3:C_2$

Intermediate fields

\(\Q(\sqrt{-11}) \), 3.1.243.1, 6.0.78594219.2, 9.1.2008387814976.4

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

Sibling fields

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.9.22.46$x^{9} + 18 x^{5} + 3$$9$$1$$22$$C_3^2 : C_6$$[2, 5/2, 17/6]_{2}$
3.9.22.46$x^{9} + 18 x^{5} + 3$$9$$1$$22$$C_3^2 : C_6$$[2, 5/2, 17/6]_{2}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$