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Properties

Label 18.8.67345067384...1408.1
Degree $18$
Signature $[8, 5]$
Discriminant $-\,2^{6}\cdot 7^{15}\cdot 53^{6}$
Root discriminant $23.95$
Ramified primes $2, 7, 53$
Class number $1$
Class group Trivial
Galois group 18T544

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-97, 607, -297, -1229, 1386, -271, -475, 1120, -1602, 786, 752, -1157, 418, 177, -194, 44, 11, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 11*x^16 + 44*x^15 - 194*x^14 + 177*x^13 + 418*x^12 - 1157*x^11 + 752*x^10 + 786*x^9 - 1602*x^8 + 1120*x^7 - 475*x^6 - 271*x^5 + 1386*x^4 - 1229*x^3 - 297*x^2 + 607*x - 97)
 
gp: K = bnfinit(x^18 - 7*x^17 + 11*x^16 + 44*x^15 - 194*x^14 + 177*x^13 + 418*x^12 - 1157*x^11 + 752*x^10 + 786*x^9 - 1602*x^8 + 1120*x^7 - 475*x^6 - 271*x^5 + 1386*x^4 - 1229*x^3 - 297*x^2 + 607*x - 97, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 11 x^{16} + 44 x^{15} - 194 x^{14} + 177 x^{13} + 418 x^{12} - 1157 x^{11} + 752 x^{10} + 786 x^{9} - 1602 x^{8} + 1120 x^{7} - 475 x^{6} - 271 x^{5} + 1386 x^{4} - 1229 x^{3} - 297 x^{2} + 607 x - 97 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-6734506738465099277161408=-\,2^{6}\cdot 7^{15}\cdot 53^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.95$
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:  $2, 7, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{15} + \frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{168815666517472486641141} a^{17} + \frac{727971145379849819734}{56271888839157495547047} a^{16} - \frac{45599240570804619461320}{168815666517472486641141} a^{15} + \frac{62645021936699082403774}{168815666517472486641141} a^{14} - \frac{11849526698608258973506}{168815666517472486641141} a^{13} + \frac{71847008648016759618917}{168815666517472486641141} a^{12} - \frac{3771795790763986506927}{18757296279719165182349} a^{11} + \frac{61928701657373240583709}{168815666517472486641141} a^{10} + \frac{192727615892082488409}{18757296279719165182349} a^{9} + \frac{17921233713084296427607}{56271888839157495547047} a^{8} - \frac{27722878879613139796337}{56271888839157495547047} a^{7} - \frac{64016466561486994796876}{168815666517472486641141} a^{6} + \frac{4452072393314473233574}{18757296279719165182349} a^{5} + \frac{72431110782741898915793}{168815666517472486641141} a^{4} - \frac{14794931013194281069414}{168815666517472486641141} a^{3} - \frac{16258292544630003681518}{56271888839157495547047} a^{2} + \frac{25719295399231323584269}{56271888839157495547047} a - \frac{285513647800199734520}{1740367696056417388053}$

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

Class group and class number

Trivial group, which has order $1$

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $12$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 385742.34864 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T544:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 9216
The 96 conjugacy class representatives for t18n544 are not computed
Character table for t18n544 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.2597.1, 9.9.17515230173.1

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 R ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
2.12.0.1$x^{12} - 26 x^{10} + 275 x^{8} - 1500 x^{6} + 4375 x^{4} - 6250 x^{2} + 7221$$1$$12$$0$$C_{12}$$[\ ]^{12}$
7Data not computed
$53$53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.12.6.1$x^{12} + 2382032 x^{6} - 418195493 x^{2} + 1418519112256$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$