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Properties

Label 18.2.48831354435...2896.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{12}\cdot 3^{45}\cdot 7^{9}$
Root discriminant $65.47$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times He_3:C_2$ (as 18T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-168143, 228402, 404766, -250188, -878454, 865368, -128073, -68634, 31968, -26240, 5310, 2124, -129, 252, -126, -24, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 24*x^15 - 126*x^14 + 252*x^13 - 129*x^12 + 2124*x^11 + 5310*x^10 - 26240*x^9 + 31968*x^8 - 68634*x^7 - 128073*x^6 + 865368*x^5 - 878454*x^4 - 250188*x^3 + 404766*x^2 + 228402*x - 168143)
 
gp: K = bnfinit(x^18 - 24*x^15 - 126*x^14 + 252*x^13 - 129*x^12 + 2124*x^11 + 5310*x^10 - 26240*x^9 + 31968*x^8 - 68634*x^7 - 128073*x^6 + 865368*x^5 - 878454*x^4 - 250188*x^3 + 404766*x^2 + 228402*x - 168143, 1)
 

Normalized defining polynomial

\( x^{18} - 24 x^{15} - 126 x^{14} + 252 x^{13} - 129 x^{12} + 2124 x^{11} + 5310 x^{10} - 26240 x^{9} + 31968 x^{8} - 68634 x^{7} - 128073 x^{6} + 865368 x^{5} - 878454 x^{4} - 250188 x^{3} + 404766 x^{2} + 228402 x - 168143 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(488313544356899506574934242512896=2^{12}\cdot 3^{45}\cdot 7^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.47$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{6}$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{6} a$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{6} a^{2}$, $\frac{1}{6} a^{12} + \frac{1}{3} a^{3} - \frac{1}{2}$, $\frac{1}{6} a^{13} + \frac{1}{3} a^{4} - \frac{1}{2} a$, $\frac{1}{18} a^{14} + \frac{1}{18} a^{13} + \frac{1}{18} a^{12} - \frac{1}{18} a^{11} - \frac{1}{18} a^{10} - \frac{1}{18} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{18} a^{5} - \frac{1}{18} a^{4} - \frac{1}{18} a^{3} + \frac{2}{9} a^{2} + \frac{2}{9} a + \frac{2}{9}$, $\frac{1}{18} a^{15} + \frac{1}{18} a^{12} + \frac{1}{18} a^{9} - \frac{7}{18} a^{6} + \frac{1}{9} a^{3} + \frac{1}{9}$, $\frac{1}{36} a^{16} - \frac{1}{36} a^{15} + \frac{1}{36} a^{13} - \frac{1}{36} a^{12} - \frac{1}{18} a^{10} + \frac{1}{18} a^{9} - \frac{1}{2} a^{8} - \frac{4}{9} a^{7} - \frac{1}{18} a^{6} - \frac{7}{36} a^{4} + \frac{7}{36} a^{3} - \frac{1}{2} a^{2} - \frac{13}{36} a - \frac{5}{36}$, $\frac{1}{1299298087933618296182626748681486512023564} a^{17} + \frac{3292966194527204390390456196005124273911}{649649043966809148091313374340743256011782} a^{16} - \frac{3274106751195968141281322643920742431519}{1299298087933618296182626748681486512023564} a^{15} - \frac{693379552208500003383129889647329101859}{44803382342538561937331956851085741793916} a^{14} + \frac{24270729874033102481234516636499159086269}{649649043966809148091313374340743256011782} a^{13} - \frac{21459582087707859282219007363596943677583}{1299298087933618296182626748681486512023564} a^{12} - \frac{7257949749446763057182370678321624811256}{108274840661134858015218895723457209335297} a^{11} + \frac{7355066705599537816563082525056537591260}{108274840661134858015218895723457209335297} a^{10} - \frac{1789996962104862048390307682946598300693}{36091613553711619338406298574485736445099} a^{9} - \frac{22940161777646855882436330284039011988665}{324824521983404574045656687170371628005891} a^{8} - \frac{12411334385767608775511730310592286158501}{649649043966809148091313374340743256011782} a^{7} - \frac{7454688095825981220235441395459947951351}{649649043966809148091313374340743256011782} a^{6} - \frac{429693358276010119669675814944038173971457}{1299298087933618296182626748681486512023564} a^{5} - \frac{15249340921985395699716693042778858308013}{649649043966809148091313374340743256011782} a^{4} + \frac{223595196031088907445899149105798327355055}{1299298087933618296182626748681486512023564} a^{3} + \frac{173273543104718573011042453451063009647597}{433099362644539432060875582893828837341188} a^{2} - \frac{13145162793897927948918122830595664986699}{36091613553711619338406298574485736445099} a - \frac{4859221862694883177879846229060133603373}{433099362644539432060875582893828837341188}$

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:  $9$
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:  \( 4590277059.47542 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times He_3:C_2$ (as 18T41):

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 $C_2\times He_3:C_2$
Character table for $C_2\times He_3:C_2$

Intermediate fields

\(\Q(\sqrt{21}) \), 3.1.243.1, 6.2.60761421.1, 9.1.2008387814976.4

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
3Data not computed
$7$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$