↑

Properties

Label 18.18.6539434525...0784.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{12}\cdot 11^{6}\cdot 37^{14}$
Root discriminant $58.55$
Ramified primes $2, 11, 37$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3\times A_4$ (as 18T31)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![783, -4491, -2838, 28402, -4663, -56919, 22166, 50687, -24800, -23129, 12782, 5671, -3454, -737, 499, 46, -36, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 36*x^16 + 46*x^15 + 499*x^14 - 737*x^13 - 3454*x^12 + 5671*x^11 + 12782*x^10 - 23129*x^9 - 24800*x^8 + 50687*x^7 + 22166*x^6 - 56919*x^5 - 4663*x^4 + 28402*x^3 - 2838*x^2 - 4491*x + 783)
 
gp: K = bnfinit(x^18 - x^17 - 36*x^16 + 46*x^15 + 499*x^14 - 737*x^13 - 3454*x^12 + 5671*x^11 + 12782*x^10 - 23129*x^9 - 24800*x^8 + 50687*x^7 + 22166*x^6 - 56919*x^5 - 4663*x^4 + 28402*x^3 - 2838*x^2 - 4491*x + 783, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 36 x^{16} + 46 x^{15} + 499 x^{14} - 737 x^{13} - 3454 x^{12} + 5671 x^{11} + 12782 x^{10} - 23129 x^{9} - 24800 x^{8} + 50687 x^{7} + 22166 x^{6} - 56919 x^{5} - 4663 x^{4} + 28402 x^{3} - 2838 x^{2} - 4491 x + 783 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(65394345253249896031868291190784=2^{12}\cdot 11^{6}\cdot 37^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.55$
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:  $2, 11, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{12} - \frac{1}{9} a^{10} - \frac{2}{9} a^{9} + \frac{2}{9} a^{8} + \frac{1}{3} a^{7} - \frac{1}{9} a^{6} + \frac{2}{9} a^{5} - \frac{4}{9} a^{4} - \frac{1}{9} a^{3} - \frac{4}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{567} a^{15} - \frac{1}{189} a^{14} - \frac{11}{567} a^{13} - \frac{10}{189} a^{12} - \frac{94}{567} a^{11} + \frac{88}{567} a^{10} - \frac{31}{81} a^{9} - \frac{29}{189} a^{8} + \frac{122}{567} a^{7} + \frac{125}{567} a^{6} - \frac{265}{567} a^{5} + \frac{206}{567} a^{4} + \frac{254}{567} a^{3} + \frac{85}{189} a^{2} + \frac{10}{63} a - \frac{8}{21}$, $\frac{1}{13041} a^{16} + \frac{5}{13041} a^{15} + \frac{103}{1863} a^{14} - \frac{1630}{13041} a^{13} + \frac{422}{13041} a^{12} + \frac{848}{13041} a^{11} + \frac{1999}{13041} a^{10} - \frac{3146}{13041} a^{9} - \frac{109}{1863} a^{8} - \frac{641}{4347} a^{7} + \frac{188}{621} a^{6} + \frac{1504}{4347} a^{5} - \frac{752}{4347} a^{4} - \frac{4895}{13041} a^{3} + \frac{269}{4347} a^{2} - \frac{7}{207} a + \frac{62}{483}$, $\frac{1}{143109104103} a^{17} - \frac{1610743}{47703034701} a^{16} - \frac{11001637}{20444157729} a^{15} + \frac{336627631}{6814719243} a^{14} + \frac{7574059727}{47703034701} a^{13} - \frac{8402859518}{143109104103} a^{12} - \frac{8924280524}{143109104103} a^{11} + \frac{18696493885}{143109104103} a^{10} + \frac{61969391767}{143109104103} a^{9} + \frac{60216934016}{143109104103} a^{8} - \frac{24202202123}{143109104103} a^{7} + \frac{65929635337}{143109104103} a^{6} - \frac{26442482648}{143109104103} a^{5} + \frac{5908615784}{143109104103} a^{4} + \frac{5516496770}{20444157729} a^{3} - \frac{8866405826}{47703034701} a^{2} + \frac{7401385843}{15901011567} a - \frac{1084354667}{5300337189}$

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:  $17$
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:  \( 17401289397.1 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times A_4$ (as 18T31):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 12 conjugacy class representatives for $S_3\times A_4$
Character table for $S_3\times A_4$

Intermediate fields

3.3.1369.1, 3.3.148.1, 6.6.226773481.1, 9.9.6075640136512.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 12 sibling: data not computed
Degree 18 sibling: 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$11$11.6.3.1$x^{6} - 22 x^{4} + 121 x^{2} - 11979$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.0.1$x^{6} + x^{2} - 2 x + 8$$1$$6$$0$$C_6$$[\ ]^{6}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$37$37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$
37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$