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Properties

Label 18.12.8718475713...0432.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,2^{30}\cdot 3^{18}\cdot 7^{12}\cdot 13^{3}\cdot 41^{3}$
Root discriminant $99.24$
Ramified primes $2, 3, 7, 13, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T926

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-140608, 616512, -316368, -1003392, 310128, 985140, 15065, -491964, -106974, 128312, 46281, -17820, -8986, 1224, 918, -32, -48, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 48*x^16 - 32*x^15 + 918*x^14 + 1224*x^13 - 8986*x^12 - 17820*x^11 + 46281*x^10 + 128312*x^9 - 106974*x^8 - 491964*x^7 + 15065*x^6 + 985140*x^5 + 310128*x^4 - 1003392*x^3 - 316368*x^2 + 616512*x - 140608)
 
gp: K = bnfinit(x^18 - 48*x^16 - 32*x^15 + 918*x^14 + 1224*x^13 - 8986*x^12 - 17820*x^11 + 46281*x^10 + 128312*x^9 - 106974*x^8 - 491964*x^7 + 15065*x^6 + 985140*x^5 + 310128*x^4 - 1003392*x^3 - 316368*x^2 + 616512*x - 140608, 1)
 

Normalized defining polynomial

\( x^{18} - 48 x^{16} - 32 x^{15} + 918 x^{14} + 1224 x^{13} - 8986 x^{12} - 17820 x^{11} + 46281 x^{10} + 128312 x^{9} - 106974 x^{8} - 491964 x^{7} + 15065 x^{6} + 985140 x^{5} + 310128 x^{4} - 1003392 x^{3} - 316368 x^{2} + 616512 x - 140608 \)

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:  \(-871847571394253513930693071110930432=-\,2^{30}\cdot 3^{18}\cdot 7^{12}\cdot 13^{3}\cdot 41^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $99.24$
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:  $2, 3, 7, 13, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{104} a^{15} + \frac{1}{26} a^{13} - \frac{3}{52} a^{12} + \frac{1}{13} a^{11} + \frac{1}{52} a^{10} + \frac{5}{52} a^{9} + \frac{2}{13} a^{8} - \frac{25}{104} a^{7} + \frac{7}{26} a^{6} + \frac{2}{13} a^{5} - \frac{11}{26} a^{4} + \frac{37}{104} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{22256} a^{16} + \frac{37}{11128} a^{15} + \frac{140}{1391} a^{14} + \frac{215}{2782} a^{13} - \frac{1141}{11128} a^{12} + \frac{77}{5564} a^{11} + \frac{1275}{11128} a^{10} - \frac{263}{2782} a^{9} - \frac{4951}{22256} a^{8} - \frac{79}{11128} a^{7} - \frac{4975}{11128} a^{6} + \frac{487}{2782} a^{5} - \frac{3271}{22256} a^{4} - \frac{5469}{11128} a^{3} + \frac{49}{107} a^{2} + \frac{27}{214} a + \frac{21}{107}$, $\frac{1}{231661920505459503074904226553696} a^{17} + \frac{37044517908136036886332253}{4455036932797298136055850510648} a^{16} + \frac{186292487668262156103047006779}{57915480126364875768726056638424} a^{15} + \frac{24318118670669990279054635186}{7239435015795609471090757079803} a^{14} - \frac{2248126483197320408203078349489}{115830960252729751537452113276848} a^{13} - \frac{1526702194057937508699984399935}{28957740063182437884363028319212} a^{12} + \frac{11062784845050999714815706980115}{115830960252729751537452113276848} a^{11} - \frac{11383222594517534582891571896533}{57915480126364875768726056638424} a^{10} - \frac{23561422131953868582189412070063}{231661920505459503074904226553696} a^{9} - \frac{3468685719098691376690880005459}{57915480126364875768726056638424} a^{8} - \frac{1720479906966418289434812231401}{115830960252729751537452113276848} a^{7} - \frac{5408831407500902788017352521115}{57915480126364875768726056638424} a^{6} + \frac{30630061535498288126148742648465}{231661920505459503074904226553696} a^{5} - \frac{45345992540629883820986517555}{2227518466398649068027925255324} a^{4} + \frac{1132098410754624783644832661673}{4455036932797298136055850510648} a^{3} - \frac{79747990088437369483288039296}{556879616599662267006981313831} a^{2} - \frac{279289383829069391886332950}{42836893584589405154383177987} a - \frac{16800520625526327217830284043}{42836893584589405154383177987}$

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:  \( 479621575786 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T926:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1119744
The 267 conjugacy class representatives for t18n926 are not computed
Character table for t18n926 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.6.6300224.1

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 ${\href{/LocalNumberField/5.9.0.1}{9} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R $18$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ R ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.12.24.284$x^{12} + 8 x^{11} - 14 x^{10} - 8 x^{9} + 8 x^{8} + 16 x^{7} - 8 x^{6} - 4 x^{4} + 16 x^{2} + 16 x + 8$$4$$3$$24$12T92$[2, 2, 2, 3, 3]^{6}$
3Data not computed
7Data not computed
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.2.2$x^{3} - 13$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.0.1$x^{6} + x^{2} - 2 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
$41$$\Q_{41}$$x + 6$$1$$1$$0$Trivial$[\ ]$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
41.3.0.1$x^{3} - x + 13$$1$$3$$0$$C_3$$[\ ]^{3}$
41.4.2.1$x^{4} + 943 x^{2} + 242064$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
41.6.0.1$x^{6} - x + 7$$1$$6$$0$$C_6$$[\ ]^{6}$