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Properties

Label 18.16.3399831347...2143.1
Degree $18$
Signature $[16, 1]$
Discriminant $-\,7^{13}\cdot 83^{6}\cdot 181^{4}$
Root discriminant $56.46$
Ramified primes $7, 83, 181$
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, -13, -981, -1079, 6374, 13795, -4380, -33368, -21488, 16738, 22549, 2699, -4873, -1126, 552, 118, -36, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 - 36*x^16 + 118*x^15 + 552*x^14 - 1126*x^13 - 4873*x^12 + 2699*x^11 + 22549*x^10 + 16738*x^9 - 21488*x^8 - 33368*x^7 - 4380*x^6 + 13795*x^5 + 6374*x^4 - 1079*x^3 - 981*x^2 - 13*x + 41)
 
gp: K = bnfinit(x^18 - 4*x^17 - 36*x^16 + 118*x^15 + 552*x^14 - 1126*x^13 - 4873*x^12 + 2699*x^11 + 22549*x^10 + 16738*x^9 - 21488*x^8 - 33368*x^7 - 4380*x^6 + 13795*x^5 + 6374*x^4 - 1079*x^3 - 981*x^2 - 13*x + 41, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} - 36 x^{16} + 118 x^{15} + 552 x^{14} - 1126 x^{13} - 4873 x^{12} + 2699 x^{11} + 22549 x^{10} + 16738 x^{9} - 21488 x^{8} - 33368 x^{7} - 4380 x^{6} + 13795 x^{5} + 6374 x^{4} - 1079 x^{3} - 981 x^{2} - 13 x + 41 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 1]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-33998313476060924767019948582143=-\,7^{13}\cdot 83^{6}\cdot 181^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.46$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} - \frac{2}{7} a^{11} - \frac{2}{7} a^{10} - \frac{3}{7} a^{9} - \frac{3}{7} a^{8} - \frac{1}{7} a^{7} + \frac{1}{7} a^{6} + \frac{2}{7} a^{3} + \frac{3}{7} a^{2} - \frac{3}{7} a - \frac{1}{7}$, $\frac{1}{7} a^{13} + \frac{1}{7} a^{11} - \frac{2}{7} a^{9} - \frac{1}{7} a^{7} + \frac{2}{7} a^{6} + \frac{2}{7} a^{4} + \frac{3}{7} a^{2} - \frac{2}{7}$, $\frac{1}{7} a^{14} + \frac{2}{7} a^{11} + \frac{3}{7} a^{9} + \frac{2}{7} a^{8} + \frac{3}{7} a^{7} - \frac{1}{7} a^{6} + \frac{2}{7} a^{5} + \frac{1}{7} a^{3} - \frac{3}{7} a^{2} + \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{49} a^{15} - \frac{3}{49} a^{14} - \frac{2}{49} a^{13} - \frac{3}{49} a^{12} + \frac{9}{49} a^{11} + \frac{13}{49} a^{10} + \frac{5}{49} a^{9} - \frac{23}{49} a^{8} + \frac{18}{49} a^{7} - \frac{11}{49} a^{6} - \frac{20}{49} a^{5} + \frac{18}{49} a^{4} - \frac{16}{49} a^{3} + \frac{3}{49} a^{2} + \frac{6}{49} a - \frac{1}{49}$, $\frac{1}{343} a^{16} - \frac{1}{343} a^{15} + \frac{20}{343} a^{14} + \frac{3}{49} a^{13} - \frac{18}{343} a^{12} + \frac{157}{343} a^{11} - \frac{74}{343} a^{10} + \frac{127}{343} a^{9} - \frac{15}{49} a^{8} - \frac{45}{343} a^{7} - \frac{19}{49} a^{6} - \frac{113}{343} a^{5} - \frac{22}{343} a^{4} + \frac{6}{343} a^{3} + \frac{47}{343} a^{2} + \frac{151}{343} a - \frac{58}{343}$, $\frac{1}{4547972845358201} a^{17} - \frac{1305377897830}{4547972845358201} a^{16} + \frac{25898908890138}{4547972845358201} a^{15} - \frac{94947641921848}{4547972845358201} a^{14} - \frac{27294548792466}{4547972845358201} a^{13} - \frac{29638996993394}{4547972845358201} a^{12} + \frac{261907006776641}{4547972845358201} a^{11} + \frac{16251407700539}{92815772354249} a^{10} - \frac{646978404577202}{4547972845358201} a^{9} - \frac{1603047754362841}{4547972845358201} a^{8} - \frac{1052786344222472}{4547972845358201} a^{7} + \frac{446187621877038}{4547972845358201} a^{6} - \frac{1562925970856337}{4547972845358201} a^{5} - \frac{1369514108515276}{4547972845358201} a^{4} - \frac{1762375339810005}{4547972845358201} a^{3} + \frac{189690942263569}{649710406479743} a^{2} - \frac{2031783544035190}{4547972845358201} a + \frac{2067030559693172}{4547972845358201}$

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:  $16$
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:  \( 11775547439.8 \) (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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ $18$ $18$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ $18$

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}$
7.6.5.5$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$$[\ ]^{2}$
181.4.2.1$x^{4} + 6335 x^{2} + 10614564$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$