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Properties

Label 18.10.2890593394...6681.1
Degree $18$
Signature $[10, 4]$
Discriminant $7^{12}\cdot 41\cdot 83^{4}\cdot 181^{4}$
Root discriminant $38.12$
Ramified primes $7, 41, 83, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T839

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41, -246, 374, 496, -2228, 1960, 1771, -4113, 1608, 1466, -1641, 284, 468, -229, -57, 50, -1, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 5*x^17 - x^16 + 50*x^15 - 57*x^14 - 229*x^13 + 468*x^12 + 284*x^11 - 1641*x^10 + 1466*x^9 + 1608*x^8 - 4113*x^7 + 1771*x^6 + 1960*x^5 - 2228*x^4 + 496*x^3 + 374*x^2 - 246*x + 41)
 
gp: K = bnfinit(x^18 - 5*x^17 - x^16 + 50*x^15 - 57*x^14 - 229*x^13 + 468*x^12 + 284*x^11 - 1641*x^10 + 1466*x^9 + 1608*x^8 - 4113*x^7 + 1771*x^6 + 1960*x^5 - 2228*x^4 + 496*x^3 + 374*x^2 - 246*x + 41, 1)
 

Normalized defining polynomial

\( x^{18} - 5 x^{17} - x^{16} + 50 x^{15} - 57 x^{14} - 229 x^{13} + 468 x^{12} + 284 x^{11} - 1641 x^{10} + 1466 x^{9} + 1608 x^{8} - 4113 x^{7} + 1771 x^{6} + 1960 x^{5} - 2228 x^{4} + 496 x^{3} + 374 x^{2} - 246 x + 41 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(28905933942693277387301036681=7^{12}\cdot 41\cdot 83^{4}\cdot 181^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.12$
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:  $7, 41, 83, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} - \frac{3}{7} a^{14} + \frac{1}{7} a^{13} - \frac{1}{7} a^{12} + \frac{1}{7} a^{11} + \frac{2}{7} a^{10} - \frac{2}{7} a^{9} + \frac{1}{7} a^{8} + \frac{2}{7} a^{7} + \frac{3}{7} a^{6} - \frac{2}{7} a^{5} + \frac{2}{7} a^{3} - \frac{1}{7} a^{2} - \frac{2}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{16} - \frac{1}{7} a^{14} + \frac{2}{7} a^{13} - \frac{2}{7} a^{12} - \frac{2}{7} a^{11} - \frac{3}{7} a^{10} + \frac{2}{7} a^{9} - \frac{2}{7} a^{8} + \frac{2}{7} a^{7} + \frac{1}{7} a^{5} + \frac{2}{7} a^{4} - \frac{2}{7} a^{3} + \frac{2}{7} a^{2} - \frac{3}{7}$, $\frac{1}{61713635800797161895077} a^{17} + \frac{2806067336117269925823}{61713635800797161895077} a^{16} + \frac{433127269518649202179}{8816233685828165985011} a^{15} - \frac{22123671693748263755694}{61713635800797161895077} a^{14} + \frac{1115258021434160347}{6389236546308847903} a^{13} - \frac{30463693533956814911138}{61713635800797161895077} a^{12} + \frac{16159228159820849865201}{61713635800797161895077} a^{11} - \frac{23992076230779090358986}{61713635800797161895077} a^{10} + \frac{19127857191031989712804}{61713635800797161895077} a^{9} - \frac{27895514819914425025352}{61713635800797161895077} a^{8} + \frac{1550496710301721101751}{4747202753907473991929} a^{7} - \frac{4810691827568216825539}{61713635800797161895077} a^{6} + \frac{5607919887863696199919}{61713635800797161895077} a^{5} + \frac{4651347126694495375663}{61713635800797161895077} a^{4} + \frac{19334134592251463938210}{61713635800797161895077} a^{3} + \frac{25741834139014414382414}{61713635800797161895077} a^{2} - \frac{25088157176609895267338}{61713635800797161895077} a + \frac{24595725707309801294359}{61713635800797161895077}$

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:  $13$
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:  \( 38592933.041 \) (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 $18$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ $18$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
41Data not computed
$83$$\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$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{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}$