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Properties

Label 18.6.35241859138...0173.2
Degree $18$
Signature $[6, 6]$
Discriminant $7^{16}\cdot 13^{9}$
Root discriminant $20.33$
Ramified primes $7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_3^2.A_4$ (as 18T92)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -7, -136, 805, 655, 623, 525, 28, 28, -259, -133, -63, -77, 0, -19, 0, 1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + x^16 - 19*x^14 - 77*x^12 - 63*x^11 - 133*x^10 - 259*x^9 + 28*x^8 + 28*x^7 + 525*x^6 + 623*x^5 + 655*x^4 + 805*x^3 - 136*x^2 - 7*x + 1)
 
gp: K = bnfinit(x^18 + x^16 - 19*x^14 - 77*x^12 - 63*x^11 - 133*x^10 - 259*x^9 + 28*x^8 + 28*x^7 + 525*x^6 + 623*x^5 + 655*x^4 + 805*x^3 - 136*x^2 - 7*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} + x^{16} - 19 x^{14} - 77 x^{12} - 63 x^{11} - 133 x^{10} - 259 x^{9} + 28 x^{8} + 28 x^{7} + 525 x^{6} + 623 x^{5} + 655 x^{4} + 805 x^{3} - 136 x^{2} - 7 x + 1 \)

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:  \(352418591388286337360173=7^{16}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.33$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{68025482879071563756800581} a^{17} - \frac{1639610700311573801868642}{68025482879071563756800581} a^{16} - \frac{1942228897916703242730535}{68025482879071563756800581} a^{15} - \frac{26740469504604491530331441}{68025482879071563756800581} a^{14} - \frac{7659368304192520523018992}{68025482879071563756800581} a^{13} - \frac{22772630496500940083888646}{68025482879071563756800581} a^{12} + \frac{7916822196434146463055245}{68025482879071563756800581} a^{11} + \frac{152833018009169065478842}{1659158119001745457482941} a^{10} + \frac{20924029086745032582484580}{68025482879071563756800581} a^{9} + \frac{14552723978111982216724695}{68025482879071563756800581} a^{8} + \frac{3873765791131644539238048}{68025482879071563756800581} a^{7} - \frac{19611793672028984322943316}{68025482879071563756800581} a^{6} - \frac{6188003308356527992879624}{68025482879071563756800581} a^{5} - \frac{29360704756097980786802973}{68025482879071563756800581} a^{4} + \frac{5324028980532457928526120}{68025482879071563756800581} a^{3} - \frac{19593041139255881162652980}{68025482879071563756800581} a^{2} + \frac{7419116926718692708525505}{68025482879071563756800581} a - \frac{1097998218370933588812434}{68025482879071563756800581}$

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:  \( 52315.3010244 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3^2.A_4$ (as 18T92):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 216
The 40 conjugacy class representatives for $C_2\times C_3^2.A_4$
Character table for $C_2\times C_3^2.A_4$ is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.2.31213.1, 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 $18$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ $18$ R $18$ R ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ $18$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ $18$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ $18$

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
7Data not computed
$13$13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.6.4.2$x^{6} - 13 x^{3} + 338$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.5.4$x^{6} + 26$$6$$1$$5$$C_6$$[\ ]_{6}$