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Properties

Label 18.4.23380963562...2672.1
Degree $18$
Signature $[4, 7]$
Discriminant $-\,2^{18}\cdot 3^{9}\cdot 7^{12}\cdot 41^{9}$
Root discriminant $81.17$
Ramified primes $2, 3, 7, 41$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T657

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![930987, 0, 3320091, 0, 3333423, 0, 1073368, 0, -20461, 0, -55632, 0, -6351, 0, 92, 0, 34, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 34*x^16 + 92*x^14 - 6351*x^12 - 55632*x^10 - 20461*x^8 + 1073368*x^6 + 3333423*x^4 + 3320091*x^2 + 930987)
 
gp: K = bnfinit(x^18 + 34*x^16 + 92*x^14 - 6351*x^12 - 55632*x^10 - 20461*x^8 + 1073368*x^6 + 3333423*x^4 + 3320091*x^2 + 930987, 1)
 

Normalized defining polynomial

\( x^{18} + 34 x^{16} + 92 x^{14} - 6351 x^{12} - 55632 x^{10} - 20461 x^{8} + 1073368 x^{6} + 3333423 x^{4} + 3320091 x^{2} + 930987 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 7]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-23380963562192407776772328774172672=-\,2^{18}\cdot 3^{9}\cdot 7^{12}\cdot 41^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.17$
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:  $2, 3, 7, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{16} a^{12} - \frac{1}{8} a^{11} - \frac{1}{16} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} + \frac{7}{16} a^{6} - \frac{3}{8} a^{5} - \frac{3}{8} a^{3} + \frac{1}{8} a^{2} + \frac{3}{8} a + \frac{3}{16}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{11} - \frac{1}{8} a^{10} - \frac{1}{4} a^{9} + \frac{7}{16} a^{7} - \frac{1}{2} a^{6} - \frac{3}{8} a^{4} + \frac{1}{8} a^{3} - \frac{3}{8} a^{2} + \frac{3}{16} a + \frac{3}{8}$, $\frac{1}{96} a^{14} - \frac{1}{48} a^{12} - \frac{7}{96} a^{10} - \frac{7}{32} a^{8} - \frac{1}{2} a^{7} + \frac{3}{32} a^{6} - \frac{5}{48} a^{4} - \frac{1}{2} a^{3} - \frac{11}{96} a^{2} - \frac{1}{2} a + \frac{3}{32}$, $\frac{1}{96} a^{15} - \frac{1}{48} a^{13} - \frac{7}{96} a^{11} - \frac{7}{32} a^{9} + \frac{3}{32} a^{7} - \frac{1}{2} a^{6} - \frac{5}{48} a^{5} - \frac{11}{96} a^{3} - \frac{1}{2} a^{2} + \frac{3}{32} a - \frac{1}{2}$, $\frac{1}{61816456020298227075648} a^{16} - \frac{39005429274194916689}{61816456020298227075648} a^{14} + \frac{1412747748008269305035}{61816456020298227075648} a^{12} - \frac{1}{8} a^{11} + \frac{52813107237428703501}{429280944585404354692} a^{10} - \frac{1}{4} a^{9} + \frac{716632993862353647335}{5151371335024852256304} a^{8} + \frac{1}{4} a^{7} - \frac{30069513789847934780101}{61816456020298227075648} a^{6} + \frac{3}{8} a^{5} - \frac{1835150675277594159461}{61816456020298227075648} a^{4} - \frac{3}{8} a^{3} + \frac{1475209954747203854881}{10302742670049704512608} a^{2} - \frac{3}{8} a - \frac{2626523849800913352707}{6868495113366469675072}$, $\frac{1}{1792677224588648585193792} a^{17} - \frac{2614691096786621044841}{1792677224588648585193792} a^{15} - \frac{8889994922041435207573}{1792677224588648585193792} a^{13} + \frac{59035921149211599095}{2575685667512426128152} a^{11} - \frac{1}{8} a^{10} + \frac{2648397244496673243449}{149389768715720715432816} a^{9} - \frac{1}{4} a^{8} + \frac{734909129461342625281043}{1792677224588648585193792} a^{7} + \frac{1}{4} a^{6} - \frac{130619434050898900567061}{1792677224588648585193792} a^{5} + \frac{3}{8} a^{4} + \frac{41559613683586084550495}{99593179143813810288544} a^{3} - \frac{3}{8} a^{2} - \frac{6919333295654956899627}{199186358287627620577088} a - \frac{3}{8}$

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:  $10$
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:  \( 10643674362.2 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T657:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 27648
The 96 conjugacy class representatives for t18n657 are not computed
Character table for t18n657 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.574470067776192.1

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 R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$$[2, 2]^{6}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
$3$3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.1$x^{6} - 6 x^{4} + 9 x^{2} - 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$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}$
41Data not computed