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Properties

Label 18.6.48255750012...7568.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 3^{18}\cdot 7^{12}\cdot 13^{3}$
Root discriminant $26.72$
Ramified primes $2, 3, 7, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T285

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-64, -192, 0, 624, 336, -1278, -251, 1416, 36, -630, -90, -54, 162, 48, -27, -4, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^16 - 4*x^15 - 27*x^14 + 48*x^13 + 162*x^12 - 54*x^11 - 90*x^10 - 630*x^9 + 36*x^8 + 1416*x^7 - 251*x^6 - 1278*x^5 + 336*x^4 + 624*x^3 - 192*x - 64)
 
gp: K = bnfinit(x^18 - 6*x^16 - 4*x^15 - 27*x^14 + 48*x^13 + 162*x^12 - 54*x^11 - 90*x^10 - 630*x^9 + 36*x^8 + 1416*x^7 - 251*x^6 - 1278*x^5 + 336*x^4 + 624*x^3 - 192*x - 64, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{16} - 4 x^{15} - 27 x^{14} + 48 x^{13} + 162 x^{12} - 54 x^{11} - 90 x^{10} - 630 x^{9} + 36 x^{8} + 1416 x^{7} - 251 x^{6} - 1278 x^{5} + 336 x^{4} + 624 x^{3} - 192 x - 64 \)

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:  \(48255750012905440263917568=2^{12}\cdot 3^{18}\cdot 7^{12}\cdot 13^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.72$
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:  $2, 3, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{6} a^{12} + \frac{1}{6} a^{9} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{10} + \frac{1}{6} a^{7} - \frac{1}{2} a^{6} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a$, $\frac{1}{12} a^{14} - \frac{1}{6} a^{11} - \frac{1}{4} a^{10} - \frac{1}{6} a^{8} - \frac{1}{2} a^{6} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{12} a^{2} - \frac{1}{2} a$, $\frac{1}{72} a^{15} - \frac{1}{12} a^{13} + \frac{1}{8} a^{11} - \frac{1}{6} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{1}{12} a^{7} + \frac{1}{4} a^{6} - \frac{35}{72} a^{3} - \frac{1}{4} a^{2} - \frac{1}{3} a + \frac{2}{9}$, $\frac{1}{144} a^{16} - \frac{1}{24} a^{14} - \frac{1}{12} a^{13} - \frac{1}{48} a^{12} + \frac{1}{6} a^{11} - \frac{5}{24} a^{10} - \frac{5}{24} a^{9} + \frac{1}{24} a^{8} - \frac{5}{24} a^{7} - \frac{1}{12} a^{6} + \frac{61}{144} a^{4} + \frac{7}{24} a^{3} - \frac{1}{6} a^{2} - \frac{1}{18} a + \frac{1}{3}$, $\frac{1}{422352958599246816} a^{17} - \frac{342093096372065}{105588239649811704} a^{16} - \frac{1199416120164557}{211176479299623408} a^{15} + \frac{509370269211797}{35196079883270568} a^{14} + \frac{11726968408474991}{140784319533082272} a^{13} - \frac{579211700579003}{35196079883270568} a^{12} + \frac{1116035193043159}{23464053255513712} a^{11} - \frac{17126795829142321}{70392159766541136} a^{10} - \frac{14477948414493343}{70392159766541136} a^{9} + \frac{2955738103912349}{23464053255513712} a^{8} + \frac{3185414339742727}{35196079883270568} a^{7} - \frac{157557905402492}{4399509985408821} a^{6} + \frac{13401022500692725}{422352958599246816} a^{5} - \frac{85608442571089537}{211176479299623408} a^{4} + \frac{17581235446386101}{105588239649811704} a^{3} + \frac{10732578818846599}{26397059912452926} a^{2} - \frac{3304626382971268}{13198529956226463} a + \frac{2099379224082601}{13198529956226463}$

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:  \( 796648.262186 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T285:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1296
The 34 conjugacy class representatives for t18n285
Character table for t18n285 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.2.31213.1, 9.5.1926650142144.4

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ $18$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.9.9.3$x^{9} + 9 x^{4} + 18 x^{3} + 54$$3$$3$$9$$(C_3^2:C_3):C_2$$[3/2, 3/2, 3/2]_{2}^{3}$
3.9.9.3$x^{9} + 9 x^{4} + 18 x^{3} + 54$$3$$3$$9$$(C_3^2:C_3):C_2$$[3/2, 3/2, 3/2]_{2}^{3}$
7Data not computed
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
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}$