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Properties

Label 18.0.17928813487...2875.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,5^{3}\cdot 7^{13}\cdot 23^{6}$
Root discriminant $15.16$
Ramified primes $5, 7, 23$
Class number $1$
Class group Trivial
Galois group $C_6\times S_4$ (as 18T61)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 12, -11, 72, -18, 241, -2, 346, 38, 286, 6, 143, -8, 50, -4, 10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 10*x^16 - 4*x^15 + 50*x^14 - 8*x^13 + 143*x^12 + 6*x^11 + 286*x^10 + 38*x^9 + 346*x^8 - 2*x^7 + 241*x^6 - 18*x^5 + 72*x^4 - 11*x^3 + 12*x^2 + 1)
 
gp: K = bnfinit(x^18 + 10*x^16 - 4*x^15 + 50*x^14 - 8*x^13 + 143*x^12 + 6*x^11 + 286*x^10 + 38*x^9 + 346*x^8 - 2*x^7 + 241*x^6 - 18*x^5 + 72*x^4 - 11*x^3 + 12*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{18} + 10 x^{16} - 4 x^{15} + 50 x^{14} - 8 x^{13} + 143 x^{12} + 6 x^{11} + 286 x^{10} + 38 x^{9} + 346 x^{8} - 2 x^{7} + 241 x^{6} - 18 x^{5} + 72 x^{4} - 11 x^{3} + 12 x^{2} + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-1792881348741312102875=-\,5^{3}\cdot 7^{13}\cdot 23^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.16$
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:  $5, 7, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3}$, $\frac{1}{3} a^{14} + \frac{1}{3} a$, $\frac{1}{21} a^{15} - \frac{1}{21} a^{14} - \frac{1}{21} a^{13} + \frac{1}{7} a^{12} + \frac{2}{21} a^{11} + \frac{2}{21} a^{10} - \frac{2}{7} a^{9} - \frac{10}{21} a^{8} + \frac{2}{7} a^{7} - \frac{8}{21} a^{6} - \frac{1}{21} a^{5} + \frac{10}{21} a^{4} + \frac{4}{21} a^{3} - \frac{10}{21} a^{2} - \frac{5}{21} a - \frac{1}{21}$, $\frac{1}{2311743} a^{16} + \frac{25129}{2311743} a^{15} - \frac{316597}{2311743} a^{14} + \frac{116420}{2311743} a^{13} + \frac{147667}{2311743} a^{12} - \frac{118060}{2311743} a^{11} + \frac{43471}{2311743} a^{10} + \frac{76528}{2311743} a^{9} - \frac{741917}{2311743} a^{8} - \frac{775216}{2311743} a^{7} - \frac{198875}{770581} a^{6} - \frac{200878}{770581} a^{5} + \frac{727024}{2311743} a^{4} + \frac{111164}{2311743} a^{3} - \frac{769487}{2311743} a^{2} + \frac{318033}{770581} a + \frac{50051}{330249}$, $\frac{1}{6935229} a^{17} - \frac{1}{6935229} a^{16} - \frac{152113}{6935229} a^{15} - \frac{66690}{770581} a^{14} - \frac{911872}{6935229} a^{13} + \frac{238409}{6935229} a^{12} - \frac{117388}{770581} a^{11} - \frac{5056}{330249} a^{10} + \frac{30034}{6935229} a^{9} + \frac{3134521}{6935229} a^{8} - \frac{342407}{2311743} a^{7} + \frac{2691589}{6935229} a^{6} + \frac{303739}{2311743} a^{5} + \frac{451406}{2311743} a^{4} - \frac{1128836}{2311743} a^{3} + \frac{3404455}{6935229} a^{2} - \frac{3052354}{6935229} a - \frac{3445643}{6935229}$

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:  $8$
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:  \( 1307.42217204 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_4$ (as 18T61):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 30 conjugacy class representatives for $C_6\times S_4$
Character table for $C_6\times S_4$ is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.1.23.1, 6.0.18515.1, 9.3.1431435383.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ R R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$5$5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.12.0.1$x^{12} - x^{3} - 2 x + 3$$1$$12$$0$$C_{12}$$[\ ]^{12}$
$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}$
$23$23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.2$x^{6} - 529 x^{2} + 48668$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.0.1$x^{6} - x + 15$$1$$6$$0$$C_6$$[\ ]^{6}$