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Properties

Label 18.6.45672975569...9733.1
Degree $18$
Signature $[6, 6]$
Discriminant $7^{12}\cdot 53^{9}$
Root discriminant $26.64$
Ramified primes $7, 53$
Class number $1$
Class group Trivial
Galois group $C_3\times S_4$ (as 18T30)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-419, -3258, -8884, 7814, -4313, -2419, 6144, -3000, 1726, -845, -326, 539, -232, 126, -38, -11, 5, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 5*x^16 - 11*x^15 - 38*x^14 + 126*x^13 - 232*x^12 + 539*x^11 - 326*x^10 - 845*x^9 + 1726*x^8 - 3000*x^7 + 6144*x^6 - 2419*x^5 - 4313*x^4 + 7814*x^3 - 8884*x^2 - 3258*x - 419)
 
gp: K = bnfinit(x^18 - 3*x^17 + 5*x^16 - 11*x^15 - 38*x^14 + 126*x^13 - 232*x^12 + 539*x^11 - 326*x^10 - 845*x^9 + 1726*x^8 - 3000*x^7 + 6144*x^6 - 2419*x^5 - 4313*x^4 + 7814*x^3 - 8884*x^2 - 3258*x - 419, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 5 x^{16} - 11 x^{15} - 38 x^{14} + 126 x^{13} - 232 x^{12} + 539 x^{11} - 326 x^{10} - 845 x^{9} + 1726 x^{8} - 3000 x^{7} + 6144 x^{6} - 2419 x^{5} - 4313 x^{4} + 7814 x^{3} - 8884 x^{2} - 3258 x - 419 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(45672975569536652017399733=7^{12}\cdot 53^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.64$
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:  $7, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4905816946456689987316036760773774742} a^{17} - \frac{299148179944178837594664638943506793}{2452908473228344993658018380386887371} a^{16} + \frac{241783069515752777663478705717319321}{2452908473228344993658018380386887371} a^{15} + \frac{84154609811662569733542029378981985}{4905816946456689987316036760773774742} a^{14} - \frac{130140465161122997662259128294471531}{2452908473228344993658018380386887371} a^{13} + \frac{1116311126137337822575691336730500139}{4905816946456689987316036760773774742} a^{12} + \frac{809071567029356604490349408153582791}{4905816946456689987316036760773774742} a^{11} - \frac{2450683565815128707844047134502905575}{4905816946456689987316036760773774742} a^{10} + \frac{1318530573955021934334298072353230491}{4905816946456689987316036760773774742} a^{9} - \frac{893379505397608096933456796184590067}{2452908473228344993658018380386887371} a^{8} + \frac{628032353672497967654130798050352323}{2452908473228344993658018380386887371} a^{7} + \frac{903212475714218013193667787710511731}{4905816946456689987316036760773774742} a^{6} - \frac{1142594823977790706144961998648889772}{2452908473228344993658018380386887371} a^{5} - \frac{1271253630145338974542516068236222137}{4905816946456689987316036760773774742} a^{4} - \frac{1110064543275105021451882042359686395}{4905816946456689987316036760773774742} a^{3} - \frac{1994947031786896059265356436577198331}{4905816946456689987316036760773774742} a^{2} - \frac{1035998992067251827943595682362592339}{4905816946456689987316036760773774742} a + \frac{631485239785139704112590466873050099}{4905816946456689987316036760773774742}$

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:  $11$
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:  \( 511646.143485 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_4$ (as 18T30):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 15 conjugacy class representatives for $C_3\times S_4$
Character table for $C_3\times S_4$

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.2597.1, 6.2.357453677.3, 9.9.17515230173.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 sibling: data not computed
Degree 18 sibling: 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$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}$
53.12.9.2$x^{12} - 106 x^{8} + 2809 x^{4} - 9528128$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$