↑

Properties

Label 18.12.2955352530...0496.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,2^{39}\cdot 3^{42}\cdot 17^{3}$
Root discriminant $93.45$
Ramified primes $2, 3, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T857

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8704, 0, -29376, 7680, 18072, -17280, 14583, 14328, -9882, -4416, -3141, 0, 1524, 288, -27, -24, -18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 18*x^16 - 24*x^15 - 27*x^14 + 288*x^13 + 1524*x^12 - 3141*x^10 - 4416*x^9 - 9882*x^8 + 14328*x^7 + 14583*x^6 - 17280*x^5 + 18072*x^4 + 7680*x^3 - 29376*x^2 + 8704)
 
gp: K = bnfinit(x^18 - 18*x^16 - 24*x^15 - 27*x^14 + 288*x^13 + 1524*x^12 - 3141*x^10 - 4416*x^9 - 9882*x^8 + 14328*x^7 + 14583*x^6 - 17280*x^5 + 18072*x^4 + 7680*x^3 - 29376*x^2 + 8704, 1)
 

Normalized defining polynomial

\( x^{18} - 18 x^{16} - 24 x^{15} - 27 x^{14} + 288 x^{13} + 1524 x^{12} - 3141 x^{10} - 4416 x^{9} - 9882 x^{8} + 14328 x^{7} + 14583 x^{6} - 17280 x^{5} + 18072 x^{4} + 7680 x^{3} - 29376 x^{2} + 8704 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[12, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-295535253012026694638815126273130496=-\,2^{39}\cdot 3^{42}\cdot 17^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $93.45$
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:  $2, 3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{5}$, $\frac{1}{128} a^{14} - \frac{1}{64} a^{12} + \frac{1}{16} a^{11} + \frac{5}{128} a^{10} - \frac{1}{4} a^{9} + \frac{1}{32} a^{8} - \frac{5}{128} a^{6} - \frac{21}{64} a^{4} - \frac{5}{16} a^{3} - \frac{41}{128} a^{2} + \frac{1}{16}$, $\frac{1}{512} a^{15} - \frac{1}{256} a^{13} - \frac{3}{64} a^{12} + \frac{5}{512} a^{11} + \frac{1}{16} a^{10} + \frac{17}{128} a^{9} - \frac{69}{512} a^{7} - \frac{1}{8} a^{6} + \frac{11}{256} a^{5} - \frac{1}{64} a^{4} + \frac{23}{512} a^{3} + \frac{1}{4} a^{2} - \frac{31}{64} a$, $\frac{1}{32768} a^{16} + \frac{27}{16384} a^{14} - \frac{3}{4096} a^{13} + \frac{3861}{32768} a^{12} - \frac{45}{1024} a^{11} + \frac{247}{8192} a^{10} - \frac{21}{128} a^{9} + \frac{2459}{32768} a^{8} + \frac{27}{512} a^{7} + \frac{1663}{16384} a^{6} + \frac{959}{4096} a^{5} - \frac{8089}{32768} a^{4} + \frac{169}{512} a^{3} + \frac{569}{2048} a^{2} + \frac{55}{512}$, $\frac{1}{17028280786114133859565568} a^{17} - \frac{17600970919690518321}{2128535098264266732445696} a^{16} - \frac{7175229317040708484837}{8514140393057066929782784} a^{15} - \frac{7803774446805530972185}{2128535098264266732445696} a^{14} - \frac{1689821608541546194112555}{17028280786114133859565568} a^{13} + \frac{229334341491892414513863}{2128535098264266732445696} a^{12} + \frac{425071856431773252951735}{4257070196528533464891392} a^{11} - \frac{64258929778564971736991}{532133774566066683111424} a^{10} + \frac{1307379728638074272414107}{17028280786114133859565568} a^{9} + \frac{330752788726160054724141}{2128535098264266732445696} a^{8} - \frac{1222732258369687325993345}{8514140393057066929782784} a^{7} - \frac{221664032598250355949599}{2128535098264266732445696} a^{6} + \frac{3255089222040613860862119}{17028280786114133859565568} a^{5} - \frac{82346830860505343888367}{2128535098264266732445696} a^{4} - \frac{149083927380289422567175}{1064267549132133366222848} a^{3} + \frac{14814721459844557043635}{133033443641516670777856} a^{2} + \frac{96469242447032387919991}{266066887283033341555712} a + \frac{4107071609265672254849}{33258360910379167694464}$

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:  $14$
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:  \( 362809636594 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T857:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 279936
The 159 conjugacy class representatives for t18n857 are not computed
Character table for t18n857 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{9})^+\), 6.6.3359232.1

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

Sibling fields

Degree 18 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 ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ $18$ R ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.9.0.1}{9} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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$2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.12.30.107$x^{12} - 6 x^{10} - 3 x^{8} + 20 x^{6} + 3 x^{4} - 14 x^{2} + 7$$4$$3$$30$12T134$[2, 2, 2, 3, 7/2, 7/2, 7/2]^{3}$
3Data not computed
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.3.2.1$x^{3} - 17$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
17.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$