Properties

Label 18.0.333...439.1
Degree $18$
Signature $[0, 9]$
Discriminant $-3.338\times 10^{28}$
Root discriminant $38.43$
Ramified primes $7, 17$
Class number $10$ (GRH)
Class group $[10]$ (GRH)
Galois group $D_{18}$ (as 18T13)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 17*x^16 - 51*x^15 + 170*x^14 - 408*x^13 + 918*x^12 - 1445*x^11 + 1870*x^10 - 1887*x^9 + 3672*x^8 - 5797*x^7 + 8058*x^6 - 8143*x^5 + 9044*x^4 - 6851*x^3 + 5916*x^2 - 2826*x + 1971)
 
gp: K = bnfinit(x^18 - 4*x^17 + 17*x^16 - 51*x^15 + 170*x^14 - 408*x^13 + 918*x^12 - 1445*x^11 + 1870*x^10 - 1887*x^9 + 3672*x^8 - 5797*x^7 + 8058*x^6 - 8143*x^5 + 9044*x^4 - 6851*x^3 + 5916*x^2 - 2826*x + 1971, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1971, -2826, 5916, -6851, 9044, -8143, 8058, -5797, 3672, -1887, 1870, -1445, 918, -408, 170, -51, 17, -4, 1]);
 

\(x^{18} - 4 x^{17} + 17 x^{16} - 51 x^{15} + 170 x^{14} - 408 x^{13} + 918 x^{12} - 1445 x^{11} + 1870 x^{10} - 1887 x^{9} + 3672 x^{8} - 5797 x^{7} + 8058 x^{6} - 8143 x^{5} + 9044 x^{4} - 6851 x^{3} + 5916 x^{2} - 2826 x + 1971\)  Toggle raw display

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

Invariants

Degree:  $18$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-33382128422738312451250336439\)\(\medspace = -\,7^{9}\cdot 17^{17}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $38.43$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $7, 17$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $2$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{15} a^{10} + \frac{1}{15} a^{9} + \frac{2}{15} a^{8} - \frac{4}{15} a^{7} - \frac{4}{15} a^{6} + \frac{2}{15} a^{5} + \frac{1}{3} a^{4} + \frac{2}{15} a^{3} - \frac{2}{15} a^{2} - \frac{2}{15} a - \frac{2}{5}$, $\frac{1}{15} a^{11} + \frac{1}{15} a^{9} - \frac{1}{15} a^{8} + \frac{1}{3} a^{7} - \frac{4}{15} a^{6} - \frac{7}{15} a^{5} + \frac{2}{15} a^{4} + \frac{1}{15} a^{3} + \frac{1}{3} a^{2} + \frac{1}{15} a + \frac{2}{5}$, $\frac{1}{15} a^{12} - \frac{2}{15} a^{9} - \frac{2}{15} a^{8} - \frac{1}{3} a^{7} + \frac{7}{15} a^{6} - \frac{1}{3} a^{5} + \frac{2}{5} a^{4} - \frac{2}{15} a^{3} - \frac{2}{15} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{45} a^{13} + \frac{1}{45} a^{12} + \frac{1}{45} a^{11} + \frac{1}{45} a^{10} + \frac{1}{9} a^{9} - \frac{7}{45} a^{8} + \frac{4}{9} a^{7} - \frac{19}{45} a^{6} - \frac{4}{9} a^{5} + \frac{1}{45} a^{4} - \frac{17}{45} a^{3} - \frac{4}{9} a^{2} - \frac{2}{15} a + \frac{1}{5}$, $\frac{1}{45} a^{14} + \frac{1}{45} a^{10} + \frac{2}{15} a^{8} + \frac{1}{15} a^{7} - \frac{4}{45} a^{6} - \frac{1}{15} a^{4} + \frac{7}{15} a^{3} + \frac{1}{9} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{405} a^{15} + \frac{2}{405} a^{14} - \frac{1}{405} a^{13} - \frac{2}{81} a^{12} + \frac{4}{135} a^{11} + \frac{2}{81} a^{10} + \frac{5}{81} a^{9} + \frac{31}{405} a^{8} - \frac{7}{15} a^{7} - \frac{106}{405} a^{6} - \frac{199}{405} a^{5} - \frac{76}{405} a^{4} + \frac{97}{405} a^{3} + \frac{22}{135} a^{2} - \frac{14}{45} a + \frac{4}{15}$, $\frac{1}{710775} a^{16} + \frac{76}{236925} a^{15} - \frac{2906}{710775} a^{14} - \frac{1363}{142155} a^{13} - \frac{6046}{710775} a^{12} + \frac{476}{142155} a^{11} + \frac{21752}{710775} a^{10} + \frac{8939}{710775} a^{9} - \frac{20287}{142155} a^{8} + \frac{313652}{710775} a^{7} + \frac{254296}{710775} a^{6} + \frac{2182}{710775} a^{5} + \frac{84391}{236925} a^{4} - \frac{12562}{142155} a^{3} - \frac{57281}{236925} a^{2} - \frac{10643}{78975} a + \frac{10432}{26325}$, $\frac{1}{1849517177664098475} a^{17} + \frac{90506097206}{369903435532819695} a^{16} - \frac{85100821688737}{369903435532819695} a^{15} + \frac{953453907851417}{97343009350742025} a^{14} - \frac{2043569804045974}{205501908629344275} a^{13} + \frac{9316541651117951}{616505725888032825} a^{12} + \frac{10174648711176034}{616505725888032825} a^{11} + \frac{32086016597163448}{1849517177664098475} a^{10} + \frac{9879548168617211}{616505725888032825} a^{9} + \frac{38086734990871574}{616505725888032825} a^{8} - \frac{21561100825771321}{123301145177606565} a^{7} - \frac{55900389910013767}{142270552128007575} a^{6} - \frac{548114351618466398}{1849517177664098475} a^{5} + \frac{405263534218173796}{1849517177664098475} a^{4} - \frac{898598524016898788}{1849517177664098475} a^{3} - \frac{152200228790276111}{616505725888032825} a^{2} - \frac{5713728458187901}{41100381725868855} a - \frac{21263061224111066}{68500636209781425}$  Toggle raw display

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

Class group and class number

$C_{10}$, which has order $10$ (assuming GRH)

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $8$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 5250337.35027 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{9}\cdot 5250337.35027 \cdot 10}{2\sqrt{33382128422738312451250336439}}\approx 2.19290095573$ (assuming GRH)

Galois group

$D_{18}$ (as 18T13):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 36
The 12 conjugacy class representatives for $D_{18}$
Character table for $D_{18}$

Intermediate fields

\(\Q(\sqrt{-119}) \), 3.1.2023.1, 6.0.487010951.1, 9.1.16748793615841.1

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

Sibling fields

Galois closure: data not computed
Degree 18 sibling: 18.2.4768875488962616064464333777.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ $18$ $18$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17Data not computed