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Properties

Label 18.12.8932638611...6947.3
Degree $18$
Signature $[12, 3]$
Discriminant $-\,7^{13}\cdot 83^{4}\cdot 181^{5}$
Root discriminant $46.13$
Ramified primes $7, 83, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T705

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1079, 6731, 8768, -43358, -29861, 90316, 43356, -77255, -31282, 29015, 12112, -4826, -2448, 387, 265, -12, -18, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 18*x^16 - 12*x^15 + 265*x^14 + 387*x^13 - 2448*x^12 - 4826*x^11 + 12112*x^10 + 29015*x^9 - 31282*x^8 - 77255*x^7 + 43356*x^6 + 90316*x^5 - 29861*x^4 - 43358*x^3 + 8768*x^2 + 6731*x - 1079)
 
gp: K = bnfinit(x^18 - x^17 - 18*x^16 - 12*x^15 + 265*x^14 + 387*x^13 - 2448*x^12 - 4826*x^11 + 12112*x^10 + 29015*x^9 - 31282*x^8 - 77255*x^7 + 43356*x^6 + 90316*x^5 - 29861*x^4 - 43358*x^3 + 8768*x^2 + 6731*x - 1079, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 18 x^{16} - 12 x^{15} + 265 x^{14} + 387 x^{13} - 2448 x^{12} - 4826 x^{11} + 12112 x^{10} + 29015 x^{9} - 31282 x^{8} - 77255 x^{7} + 43356 x^{6} + 90316 x^{5} - 29861 x^{4} - 43358 x^{3} + 8768 x^{2} + 6731 x - 1079 \)

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:  \(-893263861107131279261229596947=-\,7^{13}\cdot 83^{4}\cdot 181^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.13$
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:  $7, 83, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} - \frac{3}{7} a^{14} + \frac{3}{7} a^{13} + \frac{1}{7} a^{12} - \frac{3}{7} a^{11} - \frac{2}{7} a^{10} - \frac{3}{7} a^{9} - \frac{2}{7} a^{8} + \frac{1}{7} a^{7} - \frac{2}{7} a^{4} + \frac{2}{7} a^{3} + \frac{3}{7} a^{2} + \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{16} + \frac{1}{7} a^{14} + \frac{3}{7} a^{13} + \frac{3}{7} a^{11} - \frac{2}{7} a^{10} + \frac{3}{7} a^{9} + \frac{2}{7} a^{8} + \frac{3}{7} a^{7} - \frac{2}{7} a^{5} + \frac{3}{7} a^{4} + \frac{2}{7} a^{3} - \frac{3}{7} a^{2} + \frac{3}{7}$, $\frac{1}{27377594650949138774101458803095429147} a^{17} - \frac{750433509276124467692842281639207675}{27377594650949138774101458803095429147} a^{16} - \frac{790013521728205527762721957150164145}{27377594650949138774101458803095429147} a^{15} - \frac{6880656145965024143422619026784523260}{27377594650949138774101458803095429147} a^{14} + \frac{11272376240128467929173656172872332167}{27377594650949138774101458803095429147} a^{13} - \frac{1551999510629349441908254696012697118}{27377594650949138774101458803095429147} a^{12} + \frac{698467157861787533329904983663075324}{2105968819303779905700112215622725319} a^{11} - \frac{10373480189999210759474428048433492071}{27377594650949138774101458803095429147} a^{10} - \frac{10631098075207733890307468658306094104}{27377594650949138774101458803095429147} a^{9} + \frac{675488868749377570377895593929097599}{2105968819303779905700112215622725319} a^{8} - \frac{9078021140027815861539248544475967640}{27377594650949138774101458803095429147} a^{7} + \frac{2686963528093350886150830695097987845}{27377594650949138774101458803095429147} a^{6} - \frac{12388385777583947196850578862563146399}{27377594650949138774101458803095429147} a^{5} - \frac{7557757562854324297109239171931267869}{27377594650949138774101458803095429147} a^{4} - \frac{5365833696771117219386767812493805547}{27377594650949138774101458803095429147} a^{3} + \frac{780133360836183895703707934295662319}{3911084950135591253443065543299347021} a^{2} + \frac{12076388287245705286665670607957462137}{27377594650949138774101458803095429147} a + \frac{717035498519197878448597789126396144}{2105968819303779905700112215622725319}$

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:  \( 786798831.96 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T705:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 41472
The 64 conjugacy class representatives for t18n705 are not computed
Character table for t18n705 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.26552265046321.1

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 $18$ $18$ $18$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ $18$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
$83$$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
83.6.4.1$x^{6} + 415 x^{3} + 55112$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$C_2$$[\ ]_{2}$
181.4.2.1$x^{4} + 6335 x^{2} + 10614564$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
181.4.2.1$x^{4} + 6335 x^{2} + 10614564$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$