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Properties

Label 18.6.14571351754...7072.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{24}\cdot 7^{12}\cdot 13^{7}$
Root discriminant $25.00$
Ramified primes $2, 7, 13$
Class number $1$
Class group Trivial
Galois group 18T282

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 28, -511, 1108, 426, -1476, 1241, 274, -633, 14, 127, 236, -447, 272, -70, -16, 18, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 18*x^16 - 16*x^15 - 70*x^14 + 272*x^13 - 447*x^12 + 236*x^11 + 127*x^10 + 14*x^9 - 633*x^8 + 274*x^7 + 1241*x^6 - 1476*x^5 + 426*x^4 + 1108*x^3 - 511*x^2 + 28*x + 1)
 
gp: K = bnfinit(x^18 - 6*x^17 + 18*x^16 - 16*x^15 - 70*x^14 + 272*x^13 - 447*x^12 + 236*x^11 + 127*x^10 + 14*x^9 - 633*x^8 + 274*x^7 + 1241*x^6 - 1476*x^5 + 426*x^4 + 1108*x^3 - 511*x^2 + 28*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 18 x^{16} - 16 x^{15} - 70 x^{14} + 272 x^{13} - 447 x^{12} + 236 x^{11} + 127 x^{10} + 14 x^{9} - 633 x^{8} + 274 x^{7} + 1241 x^{6} - 1476 x^{5} + 426 x^{4} + 1108 x^{3} - 511 x^{2} + 28 x + 1 \)

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:  \(14571351754660951801987072=2^{24}\cdot 7^{12}\cdot 13^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.00$
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:  $2, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{14884578085648437965729966491} a^{17} - \frac{7004521689658771505872808408}{14884578085648437965729966491} a^{16} + \frac{5796890071046647479307231212}{14884578085648437965729966491} a^{15} - \frac{2324570156499429750986157447}{14884578085648437965729966491} a^{14} - \frac{6754735020770989085129543716}{14884578085648437965729966491} a^{13} + \frac{5203804919006808772013168530}{14884578085648437965729966491} a^{12} + \frac{2133651744738633077197722475}{14884578085648437965729966491} a^{11} + \frac{3656870719990194234447690847}{14884578085648437965729966491} a^{10} - \frac{5204821477667453402964712276}{14884578085648437965729966491} a^{9} - \frac{6892118029986187371303625420}{14884578085648437965729966491} a^{8} + \frac{6568010576179583040368850985}{14884578085648437965729966491} a^{7} + \frac{6924581126121974549276891466}{14884578085648437965729966491} a^{6} + \frac{2566197980158321102498864062}{14884578085648437965729966491} a^{5} - \frac{6065097352838819874583405924}{14884578085648437965729966491} a^{4} - \frac{2696097551479273189614887447}{14884578085648437965729966491} a^{3} - \frac{4960709534768940543139892662}{14884578085648437965729966491} a^{2} - \frac{2734556747625092800533037425}{14884578085648437965729966491} a + \frac{2900225881291580378586236003}{14884578085648437965729966491}$

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:  \( 388616.597196 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T282:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1296
The 56 conjugacy class representatives for t18n282 are not computed
Character table for t18n282 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.6.1997632.1, 9.3.1272491584.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ $18$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.6.2$x^{6} - x^{4} - 5$$2$$3$$6$$A_4\times C_2$$[2, 2]^{6}$
2.12.18.61$x^{12} - 6 x^{10} + 2 x^{8} - 4 x^{7} + 8 x^{5} + 8 x^{4} + 8 x^{3} + 8$$4$$3$$18$$C_2^2 \times A_4$$[2, 2, 2]^{6}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$13$13.3.0.1$x^{3} - 2 x + 6$$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}$
13.6.3.2$x^{6} - 338 x^{2} + 13182$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$