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Properties

Label 18.4.24144774221...1072.1
Degree $18$
Signature $[4, 7]$
Discriminant $-\,2^{6}\cdot 3^{18}\cdot 7^{15}\cdot 29^{5}$
Root discriminant $48.75$
Ramified primes $2, 3, 7, 29$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group 18T766

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-601, -9534, -5394, 35943, -12759, -31554, 25784, 6813, -12672, 567, 4701, -1977, -282, 396, -24, -75, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 75*x^15 - 24*x^14 + 396*x^13 - 282*x^12 - 1977*x^11 + 4701*x^10 + 567*x^9 - 12672*x^8 + 6813*x^7 + 25784*x^6 - 31554*x^5 - 12759*x^4 + 35943*x^3 - 5394*x^2 - 9534*x - 601)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 75*x^15 - 24*x^14 + 396*x^13 - 282*x^12 - 1977*x^11 + 4701*x^10 + 567*x^9 - 12672*x^8 + 6813*x^7 + 25784*x^6 - 31554*x^5 - 12759*x^4 + 35943*x^3 - 5394*x^2 - 9534*x - 601, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 75 x^{15} - 24 x^{14} + 396 x^{13} - 282 x^{12} - 1977 x^{11} + 4701 x^{10} + 567 x^{9} - 12672 x^{8} + 6813 x^{7} + 25784 x^{6} - 31554 x^{5} - 12759 x^{4} + 35943 x^{3} - 5394 x^{2} - 9534 x - 601 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2414477422103715329143346811072=-\,2^{6}\cdot 3^{18}\cdot 7^{15}\cdot 29^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.75$
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:  $2, 3, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7} a^{12} - \frac{3}{7} a^{11} - \frac{1}{7} a^{10} - \frac{2}{7} a^{9} + \frac{2}{7} a^{8} + \frac{1}{7} a^{6} - \frac{3}{7} a^{4} + \frac{2}{7} a^{3} + \frac{3}{7} a^{2} - \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{13} - \frac{3}{7} a^{11} + \frac{2}{7} a^{10} + \frac{3}{7} a^{9} - \frac{1}{7} a^{8} + \frac{1}{7} a^{7} + \frac{3}{7} a^{6} - \frac{3}{7} a^{5} + \frac{2}{7} a^{3} + \frac{1}{7} a^{2} - \frac{2}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{14} + \frac{3}{7} a^{7} + \frac{3}{7}$, $\frac{1}{7} a^{15} + \frac{3}{7} a^{8} + \frac{3}{7} a$, $\frac{1}{7} a^{16} + \frac{3}{7} a^{9} + \frac{3}{7} a^{2}$, $\frac{1}{73237251768763364771901380746603} a^{17} - \frac{82944242265744877171805962988}{10462464538394766395985911535229} a^{16} - \frac{2821126272898624400811085720910}{73237251768763364771901380746603} a^{15} + \frac{2130140049191788791139618620694}{73237251768763364771901380746603} a^{14} + \frac{1388924971462490177876842658801}{73237251768763364771901380746603} a^{13} - \frac{1901278927667971888431311862392}{73237251768763364771901380746603} a^{12} - \frac{4508691332304641710502548834494}{10462464538394766395985911535229} a^{11} - \frac{3568865650466972417767703424949}{10462464538394766395985911535229} a^{10} + \frac{10912295489844328363627904165086}{73237251768763364771901380746603} a^{9} - \frac{23968908243874386367990847427430}{73237251768763364771901380746603} a^{8} - \frac{16961576013218712832271478206756}{73237251768763364771901380746603} a^{7} - \frac{7983607816129647376574316845101}{73237251768763364771901380746603} a^{6} - \frac{4323682049247567919606273206904}{73237251768763364771901380746603} a^{5} - \frac{537457063983543648969016559577}{73237251768763364771901380746603} a^{4} + \frac{11518502596092420823916693625978}{73237251768763364771901380746603} a^{3} + \frac{1752530656182042690268760213596}{73237251768763364771901380746603} a^{2} - \frac{16045794235794630334218918103268}{73237251768763364771901380746603} a + \frac{4334407212420647683394493210208}{10462464538394766395985911535229}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  $10$
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:  \( 45433017.0637 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T766:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 144 conjugacy class representatives for t18n766 are not computed
Character table for t18n766 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.13632439166829.1

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 R $18$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ $18$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ $18$

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$2.6.6.1$x^{6} + x^{2} - 1$$2$$3$$6$$A_4$$[2, 2]^{3}$
2.12.0.1$x^{12} - 26 x^{10} + 275 x^{8} - 1500 x^{6} + 4375 x^{4} - 6250 x^{2} + 7221$$1$$12$$0$$C_{12}$$[\ ]^{12}$
$3$3.9.9.2$x^{9} + 18 x^{3} + 27 x + 27$$3$$3$$9$$C_3^2 : S_3 $$[3/2, 3/2]_{2}^{3}$
3.9.9.2$x^{9} + 18 x^{3} + 27 x + 27$$3$$3$$9$$C_3^2 : S_3 $$[3/2, 3/2]_{2}^{3}$
$7$7.6.5.5$x^{6} + 56$$6$$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}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.6.5.2$x^{6} + 58$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$