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Properties

Label 18.10.2748523093...7488.1
Degree $18$
Signature $[10, 4]$
Discriminant $2^{6}\cdot 7^{15}\cdot 67^{6}$
Root discriminant $25.90$
Ramified primes $2, 7, 67$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T544

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, -25, -69, 216, 142, -586, 54, 656, -355, -292, 327, 19, -131, 17, 31, -6, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 - 6*x^16 + 31*x^15 + 17*x^14 - 131*x^13 + 19*x^12 + 327*x^11 - 292*x^10 - 355*x^9 + 656*x^8 + 54*x^7 - 586*x^6 + 142*x^5 + 216*x^4 - 69*x^3 - 25*x^2 + 5*x + 1)
 
gp: K = bnfinit(x^18 - 4*x^17 - 6*x^16 + 31*x^15 + 17*x^14 - 131*x^13 + 19*x^12 + 327*x^11 - 292*x^10 - 355*x^9 + 656*x^8 + 54*x^7 - 586*x^6 + 142*x^5 + 216*x^4 - 69*x^3 - 25*x^2 + 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} - 6 x^{16} + 31 x^{15} + 17 x^{14} - 131 x^{13} + 19 x^{12} + 327 x^{11} - 292 x^{10} - 355 x^{9} + 656 x^{8} + 54 x^{7} - 586 x^{6} + 142 x^{5} + 216 x^{4} - 69 x^{3} - 25 x^{2} + 5 x + 1 \)

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:  \(27485230939984549594007488=2^{6}\cdot 7^{15}\cdot 67^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.90$
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:  $2, 7, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{80046267616631} a^{17} - \frac{21628778033990}{80046267616631} a^{16} - \frac{25545358616093}{80046267616631} a^{15} + \frac{11083315621368}{80046267616631} a^{14} + \frac{16688840783980}{80046267616631} a^{13} - \frac{16327950846266}{80046267616631} a^{12} - \frac{6212997699677}{80046267616631} a^{11} - \frac{25652049610976}{80046267616631} a^{10} + \frac{6887832553486}{80046267616631} a^{9} + \frac{17348662896155}{80046267616631} a^{8} + \frac{39356781472071}{80046267616631} a^{7} - \frac{9597379546831}{80046267616631} a^{6} - \frac{11477382643158}{80046267616631} a^{5} + \frac{28937692022947}{80046267616631} a^{4} - \frac{26404300108152}{80046267616631} a^{3} + \frac{23848248188470}{80046267616631} a^{2} + \frac{28856176101247}{80046267616631} a - \frac{1721190495527}{80046267616631}$

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:  \( 1328433.16274 \) (assuming GRH)
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

\(\Q(\zeta_{7})^+\), 3.3.469.1, 9.9.247691263309.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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
$2$2.6.6.6$x^{6} - 13 x^{4} + 7 x^{2} - 3$$2$$3$$6$$A_4\times C_2$$[2, 2, 2]^{3}$
2.12.0.1$x^{12} - 26 x^{10} + 275 x^{8} - 1500 x^{6} + 4375 x^{4} - 6250 x^{2} + 7221$$1$$12$$0$$C_{12}$$[\ ]^{12}$
$7$7.6.5.2$x^{6} - 7$$6$$1$$5$$C_6$$[\ ]_{6}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$67$67.6.3.2$x^{6} - 4489 x^{2} + 4812208$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
67.6.0.1$x^{6} + x^{2} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
67.6.3.2$x^{6} - 4489 x^{2} + 4812208$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$