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Properties

Label 18.6.77426364528...0081.1
Degree $18$
Signature $[6, 6]$
Discriminant $7^{16}\cdot 13^{12}$
Root discriminant $31.18$
Ramified primes $7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3^2.A_4$ (as 18T47)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-13, -312, 2938, 4758, -6123, -7735, 3857, 7964, -5130, -3055, 1768, 1123, -658, -154, 102, 26, -15, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 15*x^16 + 26*x^15 + 102*x^14 - 154*x^13 - 658*x^12 + 1123*x^11 + 1768*x^10 - 3055*x^9 - 5130*x^8 + 7964*x^7 + 3857*x^6 - 7735*x^5 - 6123*x^4 + 4758*x^3 + 2938*x^2 - 312*x - 13)
 
gp: K = bnfinit(x^18 - x^17 - 15*x^16 + 26*x^15 + 102*x^14 - 154*x^13 - 658*x^12 + 1123*x^11 + 1768*x^10 - 3055*x^9 - 5130*x^8 + 7964*x^7 + 3857*x^6 - 7735*x^5 - 6123*x^4 + 4758*x^3 + 2938*x^2 - 312*x - 13, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 15 x^{16} + 26 x^{15} + 102 x^{14} - 154 x^{13} - 658 x^{12} + 1123 x^{11} + 1768 x^{10} - 3055 x^{9} - 5130 x^{8} + 7964 x^{7} + 3857 x^{6} - 7735 x^{5} - 6123 x^{4} + 4758 x^{3} + 2938 x^{2} - 312 x - 13 \)

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:  \(774263645280065083180300081=7^{16}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.18$
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:  $7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{338112663860252011698043670838854693} a^{17} - \frac{81602948450234555013435282786844090}{338112663860252011698043670838854693} a^{16} - \frac{8633699923875771562629535242509573}{338112663860252011698043670838854693} a^{15} - \frac{7105348143089757181567574175679419}{338112663860252011698043670838854693} a^{14} + \frac{104200997592330146950101145493274409}{338112663860252011698043670838854693} a^{13} + \frac{115289164677322148522133303243921459}{338112663860252011698043670838854693} a^{12} + \frac{29056489088464706638566199433637079}{338112663860252011698043670838854693} a^{11} + \frac{32079082420950316765712144530900619}{338112663860252011698043670838854693} a^{10} + \frac{75719826297493149663773308945227992}{338112663860252011698043670838854693} a^{9} + \frac{155417216098245610838931302369269985}{338112663860252011698043670838854693} a^{8} + \frac{7685357536506722032780684400869469}{338112663860252011698043670838854693} a^{7} + \frac{120568041335898864016389509812593463}{338112663860252011698043670838854693} a^{6} + \frac{9092187385576444361025978891964042}{338112663860252011698043670838854693} a^{5} - \frac{147816887632187758068740591677652206}{338112663860252011698043670838854693} a^{4} - \frac{77782369846138712098233305653817631}{338112663860252011698043670838854693} a^{3} + \frac{36002298430468795252768088982074091}{338112663860252011698043670838854693} a^{2} + \frac{66649094078438490475301298378173876}{338112663860252011698043670838854693} a - \frac{148130390192888980755048288111906789}{338112663860252011698043670838854693}$

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:  \( 2770505.95053 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2.A_4$ (as 18T47):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_3^2.A_4$
Character table for $C_3^2.A_4$

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.2.405769.1, 9.9.164648481361.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$7$7.9.8.2$x^{9} - 7$$9$$1$$8$$C_9:C_3$$[\ ]_{9}^{3}$
7.9.8.2$x^{9} - 7$$9$$1$$8$$C_9:C_3$$[\ ]_{9}^{3}$
$13$13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.4.2$x^{6} - 13 x^{3} + 338$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.3.2$x^{6} - 338 x^{2} + 13182$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$