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Properties

Label 18.6.39093104223...8125.1
Degree $18$
Signature $[6, 6]$
Discriminant $5^{15}\cdot 71^{6}$
Root discriminant $15.83$
Ramified primes $5, 71$
Class number $1$
Class group Trivial
Galois group $S_3\wr C_2$ (as 18T34)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 0, -2, 5, -5, 25, -10, 10, 5, -10, -10, -25, -5, -5, -2, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^15 - 5*x^14 - 5*x^13 - 25*x^12 - 10*x^11 - 10*x^10 + 5*x^9 + 10*x^8 - 10*x^7 + 25*x^6 - 5*x^5 + 5*x^4 - 2*x^3 - 1)
 
gp: K = bnfinit(x^18 - 2*x^15 - 5*x^14 - 5*x^13 - 25*x^12 - 10*x^11 - 10*x^10 + 5*x^9 + 10*x^8 - 10*x^7 + 25*x^6 - 5*x^5 + 5*x^4 - 2*x^3 - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{15} - 5 x^{14} - 5 x^{13} - 25 x^{12} - 10 x^{11} - 10 x^{10} + 5 x^{9} + 10 x^{8} - 10 x^{7} + 25 x^{6} - 5 x^{5} + 5 x^{4} - 2 x^{3} - 1 \)

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:  \(3909310422393798828125=5^{15}\cdot 71^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $15.83$
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:  $5, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{111313} a^{16} - \frac{31795}{111313} a^{15} - \frac{22640}{111313} a^{14} - \frac{54168}{111313} a^{13} + \frac{14179}{111313} a^{12} + \frac{53485}{111313} a^{11} - \frac{12720}{111313} a^{10} - \frac{25567}{111313} a^{9} - \frac{28804}{111313} a^{8} + \frac{25567}{111313} a^{7} - \frac{12720}{111313} a^{6} - \frac{53485}{111313} a^{5} + \frac{14179}{111313} a^{4} + \frac{54168}{111313} a^{3} - \frac{22640}{111313} a^{2} + \frac{31795}{111313} a + \frac{1}{111313}$, $\frac{1}{1447069} a^{17} - \frac{5}{1447069} a^{16} + \frac{158976}{1447069} a^{15} - \frac{29910}{1447069} a^{14} + \frac{359508}{1447069} a^{13} - \frac{347694}{1447069} a^{12} - \frac{475897}{1447069} a^{11} - \frac{550803}{1447069} a^{10} - \frac{441460}{1447069} a^{9} + \frac{229771}{1447069} a^{8} + \frac{399936}{1447069} a^{7} - \frac{22156}{1447069} a^{6} - \frac{190522}{1447069} a^{5} - \frac{458324}{1447069} a^{4} + \frac{77283}{1447069} a^{3} - \frac{389199}{1447069} a^{2} + \frac{708889}{1447069} a + \frac{31790}{1447069}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 6472.48552936 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\wr C_2$ (as 18T34):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 9 conjugacy class representatives for $S_3\wr C_2$
Character table for $S_3\wr C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), 6.4.221875.1 x2, 9.3.27961796875.1

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

Sibling fields

Degree 6 siblings: 6.4.221875.1, 6.0.44738875.1
Degree 9 sibling: 9.3.27961796875.1
Degree 12 siblings: Deg 12, 12.6.3495224609375.1, Deg 12, Deg 12, Deg 12, Deg 12
Degree 18 siblings: Deg 18, Deg 18

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/3.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$5$5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
71Data not computed