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Properties

Label 18.6.74140900471...6601.1
Degree $18$
Signature $[6, 6]$
Discriminant $7^{13}\cdot 83^{5}\cdot 181^{5}$
Root discriminant $58.96$
Ramified primes $7, 83, 181$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 18T839

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![403481, -853458, 1135381, -753228, 156071, 115392, -160451, 63304, -3864, -14085, 7410, -2690, 800, 376, -235, 50, -1, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 - x^16 + 50*x^15 - 235*x^14 + 376*x^13 + 800*x^12 - 2690*x^11 + 7410*x^10 - 14085*x^9 - 3864*x^8 + 63304*x^7 - 160451*x^6 + 115392*x^5 + 156071*x^4 - 753228*x^3 + 1135381*x^2 - 853458*x + 403481)
 
gp: K = bnfinit(x^18 - 5*x^17 - x^16 + 50*x^15 - 235*x^14 + 376*x^13 + 800*x^12 - 2690*x^11 + 7410*x^10 - 14085*x^9 - 3864*x^8 + 63304*x^7 - 160451*x^6 + 115392*x^5 + 156071*x^4 - 753228*x^3 + 1135381*x^2 - 853458*x + 403481, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} - x^{16} + 50 x^{15} - 235 x^{14} + 376 x^{13} + 800 x^{12} - 2690 x^{11} + 7410 x^{10} - 14085 x^{9} - 3864 x^{8} + 63304 x^{7} - 160451 x^{6} + 115392 x^{5} + 156071 x^{4} - 753228 x^{3} + 1135381 x^{2} - 853458 x + 403481 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(74140900471891896178682056546601=7^{13}\cdot 83^{5}\cdot 181^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.96$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} + \frac{3}{7} a^{14} - \frac{3}{7} a^{13} + \frac{3}{7} a^{12} - \frac{3}{7} a^{11} - \frac{3}{7} a^{10} - \frac{3}{7} a^{9} - \frac{1}{7} a^{8} + \frac{1}{7} a^{5} - \frac{3}{7} a^{4} + \frac{2}{7} a^{3} - \frac{1}{7} a^{2} - \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{16} + \frac{2}{7} a^{14} - \frac{2}{7} a^{13} + \frac{2}{7} a^{12} - \frac{1}{7} a^{11} - \frac{1}{7} a^{10} + \frac{1}{7} a^{9} + \frac{3}{7} a^{8} + \frac{1}{7} a^{6} + \frac{1}{7} a^{5} - \frac{3}{7} a^{4} + \frac{1}{7} a^{2} - \frac{3}{7}$, $\frac{1}{168745009901793037170579668277639317661892162283509} a^{17} + \frac{2573586743641439973984400685036829148504865484612}{168745009901793037170579668277639317661892162283509} a^{16} + \frac{5415074658732831250548531521645363940401893716813}{168745009901793037170579668277639317661892162283509} a^{15} + \frac{61804259185484860851330097577688866842116565093570}{168745009901793037170579668277639317661892162283509} a^{14} + \frac{83349631615140175233749108567291566769991910106486}{168745009901793037170579668277639317661892162283509} a^{13} + \frac{6542676347236168038360575812182352908010958411948}{168745009901793037170579668277639317661892162283509} a^{12} - \frac{13488661850747916388716645026976311539621407339625}{168745009901793037170579668277639317661892162283509} a^{11} + \frac{84078466221513715538878438138913912075527670859214}{168745009901793037170579668277639317661892162283509} a^{10} + \frac{53203993860239962559520769897944025319176749721336}{168745009901793037170579668277639317661892162283509} a^{9} + \frac{2235872084559485490023765365263447508786705760552}{168745009901793037170579668277639317661892162283509} a^{8} + \frac{38115497664144305774880119665529475667802962197717}{168745009901793037170579668277639317661892162283509} a^{7} - \frac{71207749266218650544391821902048484484719676998849}{168745009901793037170579668277639317661892162283509} a^{6} + \frac{125360440721032026814099533616392003062219169175}{587961706974888631256375150793168354222620774507} a^{5} - \frac{78945270477380274183137365935440549892303122167219}{168745009901793037170579668277639317661892162283509} a^{4} - \frac{21208825676640135448954795956308149923469949238532}{168745009901793037170579668277639317661892162283509} a^{3} + \frac{6746437626368553319072482222427945460267906780975}{168745009901793037170579668277639317661892162283509} a^{2} + \frac{29136135076577941010467005718378550068406096587737}{168745009901793037170579668277639317661892162283509} a - \frac{1026668918947934242003744868592607482720676539818}{4115731948824220418794626055552178479558345421549}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 725773882.856 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T839:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 165888
The 192 conjugacy class representatives for t18n839 are not computed
Character table for t18n839 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ $18$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ $18$

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}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
83Data not computed
$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}$
181.2.1.2$x^{2} + 362$$2$$1$$1$$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$$[\ ]_{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.4.3.3$x^{4} + 362$$4$$1$$3$$C_4$$[\ ]_{4}$