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Properties

Label 18.10.1432597605...7673.1
Degree $18$
Signature $[10, 4]$
Discriminant $7^{13}\cdot 52879^{3}$
Root discriminant $24.98$
Ramified primes $7, 52879$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T765

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 15, 3, -53, 83, -401, 340, 34, -848, 889, 101, -570, 240, 30, -52, 31, -9, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 9*x^16 + 31*x^15 - 52*x^14 + 30*x^13 + 240*x^12 - 570*x^11 + 101*x^10 + 889*x^9 - 848*x^8 + 34*x^7 + 340*x^6 - 401*x^5 + 83*x^4 - 53*x^3 + 3*x^2 + 15*x + 1)
 
gp: K = bnfinit(x^18 - 2*x^17 - 9*x^16 + 31*x^15 - 52*x^14 + 30*x^13 + 240*x^12 - 570*x^11 + 101*x^10 + 889*x^9 - 848*x^8 + 34*x^7 + 340*x^6 - 401*x^5 + 83*x^4 - 53*x^3 + 3*x^2 + 15*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 9 x^{16} + 31 x^{15} - 52 x^{14} + 30 x^{13} + 240 x^{12} - 570 x^{11} + 101 x^{10} + 889 x^{9} - 848 x^{8} + 34 x^{7} + 340 x^{6} - 401 x^{5} + 83 x^{4} - 53 x^{3} + 3 x^{2} + 15 x + 1 \)

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:  \(14325976053911690018397673=7^{13}\cdot 52879^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.98$
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:  $7, 52879$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{93458516912454310730723} a^{17} - \frac{25291226757842102441355}{93458516912454310730723} a^{16} - \frac{15295719119965496093443}{93458516912454310730723} a^{15} + \frac{45648197402971961632053}{93458516912454310730723} a^{14} + \frac{12051534945780834405725}{93458516912454310730723} a^{13} - \frac{24866988682711612128379}{93458516912454310730723} a^{12} + \frac{34806232513405324051327}{93458516912454310730723} a^{11} - \frac{26813754468437557237265}{93458516912454310730723} a^{10} + \frac{33313457184257670615893}{93458516912454310730723} a^{9} + \frac{3816619159754638733854}{93458516912454310730723} a^{8} - \frac{32569261085477350134589}{93458516912454310730723} a^{7} + \frac{30983157910450091540795}{93458516912454310730723} a^{6} + \frac{21634296323883502012111}{93458516912454310730723} a^{5} - \frac{23310686830088309798517}{93458516912454310730723} a^{4} + \frac{9457161901761847963388}{93458516912454310730723} a^{3} - \frac{6748346750375050016376}{93458516912454310730723} a^{2} - \frac{33155279644786692841079}{93458516912454310730723} a - \frac{41320890368931659618195}{93458516912454310730723}$

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:  \( 981664.03222 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T765:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 110 conjugacy class representatives for t18n765 are not computed
Character table for t18n765 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.7.6221161471.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ $18$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ $18$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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
$7$7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
52879Data not computed