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Properties

Label 18.12.3862757309...8323.1
Degree $18$
Signature $[12, 3]$
Discriminant $-\,3^{18}\cdot 7^{14}\cdot 43^{5}$
Root discriminant $38.74$
Ramified primes $3, 7, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T282

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7267, -15093, 59598, 152048, 99588, -28251, -81698, -36273, 16047, 12458, -4620, -4167, -69, 600, 207, -25, -27, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 27*x^16 - 25*x^15 + 207*x^14 + 600*x^13 - 69*x^12 - 4167*x^11 - 4620*x^10 + 12458*x^9 + 16047*x^8 - 36273*x^7 - 81698*x^6 - 28251*x^5 + 99588*x^4 + 152048*x^3 + 59598*x^2 - 15093*x - 7267)
 
gp: K = bnfinit(x^18 - 27*x^16 - 25*x^15 + 207*x^14 + 600*x^13 - 69*x^12 - 4167*x^11 - 4620*x^10 + 12458*x^9 + 16047*x^8 - 36273*x^7 - 81698*x^6 - 28251*x^5 + 99588*x^4 + 152048*x^3 + 59598*x^2 - 15093*x - 7267, 1)
 

Normalized defining polynomial

\( x^{18} - 27 x^{16} - 25 x^{15} + 207 x^{14} + 600 x^{13} - 69 x^{12} - 4167 x^{11} - 4620 x^{10} + 12458 x^{9} + 16047 x^{8} - 36273 x^{7} - 81698 x^{6} - 28251 x^{5} + 99588 x^{4} + 152048 x^{3} + 59598 x^{2} - 15093 x - 7267 \)

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:  \(-38627573098228816471588288323=-\,3^{18}\cdot 7^{14}\cdot 43^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.74$
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:  $3, 7, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{41199977727433375353930421894746321397} a^{17} - \frac{730245831734160125745571261753726535}{3169229055956413488763878607288178569} a^{16} + \frac{19279557090840202227759365419190989591}{41199977727433375353930421894746321397} a^{15} + \frac{640776957187703145694467380107156141}{1420688887152875012204497306715390393} a^{14} + \frac{16946171033892423660947864866115814725}{41199977727433375353930421894746321397} a^{13} - \frac{3281134393722362180812474912154771731}{41199977727433375353930421894746321397} a^{12} + \frac{16013634858234045317750724816513601030}{41199977727433375353930421894746321397} a^{11} + \frac{9984276131135711014765636519216136406}{41199977727433375353930421894746321397} a^{10} - \frac{2330648079447544130910248651730919063}{41199977727433375353930421894746321397} a^{9} + \frac{11059830685759426075554696846708340381}{41199977727433375353930421894746321397} a^{8} + \frac{19974407528173509977054596993519328492}{41199977727433375353930421894746321397} a^{7} - \frac{20367095780434991990471049141763362764}{41199977727433375353930421894746321397} a^{6} + \frac{3966278205917034452280147489627170559}{41199977727433375353930421894746321397} a^{5} + \frac{10611472328574325561407514885508948473}{41199977727433375353930421894746321397} a^{4} + \frac{8631887963624084881019153790298265858}{41199977727433375353930421894746321397} a^{3} + \frac{1227243834486799203832605676724180315}{3169229055956413488763878607288178569} a^{2} - \frac{10366592017161278458494862837576895638}{41199977727433375353930421894746321397} a - \frac{1102409572555314847631597068422303470}{3169229055956413488763878607288178569}$

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:  \( 88012551.4345 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T282:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1296
The 56 conjugacy class representatives for t18n282 are not computed
Character table for t18n282 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.4.103243.1, 9.9.29971914410781.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R $18$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ R ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
3Data not computed
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$43$43.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
43.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
43.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
43.3.0.1$x^{3} - x + 10$$1$$3$$0$$C_3$$[\ ]^{3}$
43.6.5.6$x^{6} + 2539107$$6$$1$$5$$C_6$$[\ ]_{6}$