↑

Properties

Label 18.6.12150082333...7549.1
Degree $18$
Signature $[6, 6]$
Discriminant $7^{12}\cdot 23^{6}\cdot 181^{3}$
Root discriminant $24.75$
Ramified primes $7, 23, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times S_3\times A_4$ (as 18T60)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 12, 17, -44, -82, 95, 140, -150, -226, 118, 178, -67, -60, 26, 22, -4, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 4*x^16 + 22*x^15 + 26*x^14 - 60*x^13 - 67*x^12 + 178*x^11 + 118*x^10 - 226*x^9 - 150*x^8 + 140*x^7 + 95*x^6 - 82*x^5 - 44*x^4 + 17*x^3 + 12*x^2 - 1)
 
gp: K = bnfinit(x^18 - 2*x^17 - 4*x^16 + 22*x^15 + 26*x^14 - 60*x^13 - 67*x^12 + 178*x^11 + 118*x^10 - 226*x^9 - 150*x^8 + 140*x^7 + 95*x^6 - 82*x^5 - 44*x^4 + 17*x^3 + 12*x^2 - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 4 x^{16} + 22 x^{15} + 26 x^{14} - 60 x^{13} - 67 x^{12} + 178 x^{11} + 118 x^{10} - 226 x^{9} - 150 x^{8} + 140 x^{7} + 95 x^{6} - 82 x^{5} - 44 x^{4} + 17 x^{3} + 12 x^{2} - 1 \)

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:  \(12150082333447607737387549=7^{12}\cdot 23^{6}\cdot 181^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.75$
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:  $7, 23, 181$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{820803} a^{16} + \frac{32325}{273601} a^{15} + \frac{11266}{273601} a^{14} - \frac{25545}{273601} a^{13} + \frac{193004}{820803} a^{12} + \frac{38616}{273601} a^{11} + \frac{78731}{820803} a^{10} - \frac{231833}{820803} a^{9} + \frac{34289}{820803} a^{8} - \frac{28556}{820803} a^{7} - \frac{77348}{820803} a^{6} - \frac{215830}{820803} a^{5} + \frac{225355}{820803} a^{4} + \frac{86230}{820803} a^{3} + \frac{99203}{273601} a^{2} + \frac{3728}{820803} a + \frac{318269}{820803}$, $\frac{1}{3897993447} a^{17} + \frac{278}{3897993447} a^{16} - \frac{477738050}{3897993447} a^{15} + \frac{616345157}{3897993447} a^{14} - \frac{68862938}{3897993447} a^{13} - \frac{121672973}{3897993447} a^{12} - \frac{644705017}{1299331149} a^{11} + \frac{359723401}{3897993447} a^{10} - \frac{1312372042}{3897993447} a^{9} + \frac{559065373}{3897993447} a^{8} - \frac{1243222052}{3897993447} a^{7} + \frac{550649876}{1299331149} a^{6} - \frac{1387376086}{3897993447} a^{5} + \frac{1722750937}{3897993447} a^{4} + \frac{1589727584}{3897993447} a^{3} + \frac{1778686738}{3897993447} a^{2} + \frac{779380018}{3897993447} a + \frac{1323485428}{3897993447}$

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:  \( 341295.043533 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_3\times A_4$ (as 18T60):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 24 conjugacy class representatives for $C_2\times S_3\times A_4$
Character table for $C_2\times S_3\times A_4$ is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.1.23.1, 6.6.434581.1, 9.3.1431435383.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$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$$[\ ]_{3}^{2}$
$23$23.6.0.1$x^{6} - x + 15$$1$$6$$0$$C_6$$[\ ]^{6}$
23.12.6.1$x^{12} + 365010 x^{6} - 6436343 x^{2} + 33308075025$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
181Data not computed