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Properties

Label 18.0.45738215713...9632.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{31}\cdot 7^{10}$
Root discriminant $39.11$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times S_3^2$ (as 18T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2779, 10773, 19005, 24864, 25599, 21609, 16185, 8448, 7878, 1234, 3060, -438, 963, -237, 189, -42, 21, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 21*x^16 - 42*x^15 + 189*x^14 - 237*x^13 + 963*x^12 - 438*x^11 + 3060*x^10 + 1234*x^9 + 7878*x^8 + 8448*x^7 + 16185*x^6 + 21609*x^5 + 25599*x^4 + 24864*x^3 + 19005*x^2 + 10773*x + 2779)
 
gp: K = bnfinit(x^18 - 3*x^17 + 21*x^16 - 42*x^15 + 189*x^14 - 237*x^13 + 963*x^12 - 438*x^11 + 3060*x^10 + 1234*x^9 + 7878*x^8 + 8448*x^7 + 16185*x^6 + 21609*x^5 + 25599*x^4 + 24864*x^3 + 19005*x^2 + 10773*x + 2779, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 21 x^{16} - 42 x^{15} + 189 x^{14} - 237 x^{13} + 963 x^{12} - 438 x^{11} + 3060 x^{10} + 1234 x^{9} + 7878 x^{8} + 8448 x^{7} + 16185 x^{6} + 21609 x^{5} + 25599 x^{4} + 24864 x^{3} + 19005 x^{2} + 10773 x + 2779 \)

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:  \(-45738215713271605763192389632=-\,2^{18}\cdot 3^{31}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.11$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{4} a^{12} - \frac{1}{4}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{12} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{3}{8} a + \frac{3}{8}$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} - \frac{1}{8} a^{9} + \frac{1}{8} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{1}{8} a^{3} + \frac{5}{16} a^{2} - \frac{1}{4} a + \frac{5}{16}$, $\frac{1}{16} a^{15} - \frac{1}{16} a^{13} - \frac{1}{8} a^{12} - \frac{1}{8} a^{11} - \frac{1}{8} a^{10} + \frac{1}{8} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} + \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{5}{16} a^{3} - \frac{1}{4} a^{2} + \frac{5}{16} a$, $\frac{1}{32} a^{16} - \frac{1}{32} a^{15} - \frac{1}{32} a^{13} + \frac{3}{32} a^{12} - \frac{1}{16} a^{11} + \frac{1}{16} a^{10} + \frac{1}{16} a^{8} - \frac{3}{16} a^{7} + \frac{1}{8} a^{6} + \frac{1}{16} a^{5} + \frac{5}{32} a^{4} - \frac{15}{32} a^{3} - \frac{1}{16} a^{2} + \frac{15}{32} a + \frac{9}{32}$, $\frac{1}{51928344662587647099349568} a^{17} + \frac{79853496839634999767961}{25964172331293823549674784} a^{16} + \frac{449652297059929739708735}{51928344662587647099349568} a^{15} - \frac{1370916030407153671297439}{51928344662587647099349568} a^{14} - \frac{764408652706996870070721}{25964172331293823549674784} a^{13} + \frac{5169680019038892210281193}{51928344662587647099349568} a^{12} - \frac{434188476279296489700327}{12982086165646911774837392} a^{11} - \frac{2658747435543822782741365}{25964172331293823549674784} a^{10} - \frac{2493515306641205598779475}{25964172331293823549674784} a^{9} - \frac{1857489278453344815236249}{12982086165646911774837392} a^{8} + \frac{987777772945211740310317}{25964172331293823549674784} a^{7} + \frac{1140351042730773709090177}{25964172331293823549674784} a^{6} + \frac{23397329887772947471784435}{51928344662587647099349568} a^{5} - \frac{304970219708642287967277}{1622760770705863971854674} a^{4} + \frac{991835891969724326162903}{51928344662587647099349568} a^{3} + \frac{7644952656320055694203291}{51928344662587647099349568} a^{2} + \frac{329067282236211059681635}{811380385352931985927337} a - \frac{1204261673993547533254091}{51928344662587647099349568}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( -\frac{20216776176489}{31644527977671328} a^{17} + \frac{80572660509387}{31644527977671328} a^{16} - \frac{252878392018891}{15822263988835664} a^{15} + \frac{1336997463921951}{31644527977671328} a^{14} - \frac{5115633662243541}{31644527977671328} a^{13} + \frac{4777757749037495}{15822263988835664} a^{12} - \frac{14166216300123573}{15822263988835664} a^{11} + \frac{2126187456999285}{1977782998604458} a^{10} - \frac{45760823395852335}{15822263988835664} a^{9} + \frac{25335028679956983}{15822263988835664} a^{8} - \frac{6199096917359691}{988891499302229} a^{7} - \frac{8851738712061425}{15822263988835664} a^{6} - \frac{306864236180407605}{31644527977671328} a^{5} - \frac{220025757206466603}{31644527977671328} a^{4} - \frac{84933795684126571}{7911131994417832} a^{3} - \frac{277071506800580037}{31644527977671328} a^{2} - \frac{206087895111114219}{31644527977671328} a - \frac{19771420435579315}{7911131994417832} \) (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:  \( 118048197.30834381 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.1512.1, 6.0.964467.3, 6.0.6858432.1

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

Sibling fields

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$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$$[\ ]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.2$x^{4} - 2 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
3Data not computed
$7$7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.6.5.3$x^{6} - 112$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$