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Properties

Label 18.6.60874624883...9904.2
Degree $18$
Signature $[6, 6]$
Discriminant $2^{42}\cdot 7^{12}$
Root discriminant $18.44$
Ramified primes $2, 7$
Class number $1$
Class group Trivial
Galois group $C_3\times C_3:S_3.C_2$ (as 18T44)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, 16, 104, -176, -152, 336, 0, -208, 106, -76, 98, 12, -103, 70, -27, 10, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 2*x^16 + 10*x^15 - 27*x^14 + 70*x^13 - 103*x^12 + 12*x^11 + 98*x^10 - 76*x^9 + 106*x^8 - 208*x^7 + 336*x^5 - 152*x^4 - 176*x^3 + 104*x^2 + 16*x - 8)
 
gp: K = bnfinit(x^18 - 4*x^17 + 2*x^16 + 10*x^15 - 27*x^14 + 70*x^13 - 103*x^12 + 12*x^11 + 98*x^10 - 76*x^9 + 106*x^8 - 208*x^7 + 336*x^5 - 152*x^4 - 176*x^3 + 104*x^2 + 16*x - 8, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 2 x^{16} + 10 x^{15} - 27 x^{14} + 70 x^{13} - 103 x^{12} + 12 x^{11} + 98 x^{10} - 76 x^{9} + 106 x^{8} - 208 x^{7} + 336 x^{5} - 152 x^{4} - 176 x^{3} + 104 x^{2} + 16 x - 8 \)

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:  \(60874624883546499579904=2^{42}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.44$
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:  $2, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{284} a^{16} - \frac{1}{142} a^{15} - \frac{17}{284} a^{14} + \frac{3}{142} a^{13} + \frac{27}{284} a^{12} + \frac{17}{142} a^{11} - \frac{7}{142} a^{10} - \frac{25}{71} a^{9} - \frac{105}{284} a^{8} - \frac{16}{71} a^{7} - \frac{20}{71} a^{6} + \frac{6}{71} a^{5} - \frac{15}{142} a^{4} - \frac{21}{71} a^{3} - \frac{3}{71} a^{2} - \frac{19}{71} a + \frac{33}{71}$, $\frac{1}{20486074359748} a^{17} - \frac{35091258163}{20486074359748} a^{16} - \frac{273291610130}{5121518589937} a^{15} - \frac{182355469771}{20486074359748} a^{14} + \frac{4754566958811}{20486074359748} a^{13} + \frac{4319976871985}{20486074359748} a^{12} + \frac{472633929847}{20486074359748} a^{11} - \frac{391644644849}{5121518589937} a^{10} - \frac{1937146799521}{10243037179874} a^{9} + \frac{2438595045927}{20486074359748} a^{8} - \frac{3924258540939}{10243037179874} a^{7} - \frac{1580116379967}{10243037179874} a^{6} + \frac{1498075054949}{5121518589937} a^{5} - \frac{1507215488333}{5121518589937} a^{4} - \frac{5607900391}{5121518589937} a^{3} - \frac{1386251139825}{5121518589937} a^{2} - \frac{1505249853248}{5121518589937} a - \frac{1476916335330}{5121518589937}$

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:  $11$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 26652.5198107 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_3:S_3.C_2$ (as 18T44):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 18 conjugacy class representatives for $C_3\times C_3:S_3.C_2$
Character table for $C_3\times C_3:S_3.C_2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{7})^+\), 6.6.1229312.1, 6.2.802816.1

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

Sibling fields

Degree 12 siblings: data not computed
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 R ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }$

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$2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.12.33.344$x^{12} + 4 x^{10} + 10 x^{8} - 8 x^{6} + 8 x^{4} + 32 x^{2} + 8$$4$$3$$33$$C_{12}$$[3, 4]^{3}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$