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Properties

Label 18.6.24144774221...1072.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{6}\cdot 3^{18}\cdot 7^{15}\cdot 29^{5}$
Root discriminant $48.75$
Ramified primes $2, 3, 7, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T766

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-48539, 96579, 37485, -202983, 68934, 175326, -137394, -46203, 79359, -8259, -19848, 5445, 2972, -1230, -219, 147, -6, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 6*x^16 + 147*x^15 - 219*x^14 - 1230*x^13 + 2972*x^12 + 5445*x^11 - 19848*x^10 - 8259*x^9 + 79359*x^8 - 46203*x^7 - 137394*x^6 + 175326*x^5 + 68934*x^4 - 202983*x^3 + 37485*x^2 + 96579*x - 48539)
 
gp: K = bnfinit(x^18 - 6*x^17 - 6*x^16 + 147*x^15 - 219*x^14 - 1230*x^13 + 2972*x^12 + 5445*x^11 - 19848*x^10 - 8259*x^9 + 79359*x^8 - 46203*x^7 - 137394*x^6 + 175326*x^5 + 68934*x^4 - 202983*x^3 + 37485*x^2 + 96579*x - 48539, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 6 x^{16} + 147 x^{15} - 219 x^{14} - 1230 x^{13} + 2972 x^{12} + 5445 x^{11} - 19848 x^{10} - 8259 x^{9} + 79359 x^{8} - 46203 x^{7} - 137394 x^{6} + 175326 x^{5} + 68934 x^{4} - 202983 x^{3} + 37485 x^{2} + 96579 x - 48539 \)

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:  \(2414477422103715329143346811072=2^{6}\cdot 3^{18}\cdot 7^{15}\cdot 29^{5}\)
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{26} a^{16} + \frac{5}{26} a^{15} - \frac{5}{26} a^{14} - \frac{9}{26} a^{13} + \frac{2}{13} a^{12} - \frac{11}{26} a^{11} + \frac{9}{26} a^{10} + \frac{1}{13} a^{9} + \frac{7}{26} a^{8} + \frac{2}{13} a^{7} + \frac{11}{26} a^{6} - \frac{5}{26} a^{5} + \frac{2}{13} a^{4} + \frac{5}{13} a^{3} - \frac{7}{26} a^{2} - \frac{7}{26} a - \frac{5}{26}$, $\frac{1}{3712086711130747643481278664513647841326} a^{17} + \frac{61143475886316037169556773204211032713}{3712086711130747643481278664513647841326} a^{16} - \frac{761378304999432563640291881818320212723}{3712086711130747643481278664513647841326} a^{15} + \frac{951721555281947884292557889793409952307}{3712086711130747643481278664513647841326} a^{14} + \frac{862520522260627282920667410625444709964}{1856043355565373821740639332256823920663} a^{13} + \frac{1086620790402319723297527477558516481141}{3712086711130747643481278664513647841326} a^{12} + \frac{1516437810598027568749315977717832302395}{3712086711130747643481278664513647841326} a^{11} + \frac{190227373262536578478196731925571811504}{1856043355565373821740639332256823920663} a^{10} - \frac{1691476227024181566280563846968491879045}{3712086711130747643481278664513647841326} a^{9} + \frac{624948726871265358499969000434648887481}{1856043355565373821740639332256823920663} a^{8} + \frac{227676109944207703230212094088107289013}{3712086711130747643481278664513647841326} a^{7} - \frac{531558024910737686871903153989953467515}{3712086711130747643481278664513647841326} a^{6} + \frac{170671937705484777326664090033487596721}{1856043355565373821740639332256823920663} a^{5} - \frac{32107562076124069363293622611438375470}{1856043355565373821740639332256823920663} a^{4} - \frac{979269702720576856035171083754555432751}{3712086711130747643481278664513647841326} a^{3} + \frac{801082580229823339796108157733814953827}{3712086711130747643481278664513647841326} a^{2} - \frac{907045609237150815898228129375713405841}{3712086711130747643481278664513647841326} a - \frac{159651226453855182571527465385331672314}{1856043355565373821740639332256823920663}$

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:  $11$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 307000319.813 \) (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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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$2.6.6.4$x^{6} + x^{2} + 1$$2$$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}$
3Data not computed
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.12.11.2$x^{12} + 56$$12$$1$$11$$D_4 \times C_3$$[\ ]_{12}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$