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Properties

Label 18.0.22701208004...6003.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{33}\cdot 17^{12}\cdot 307^{12}$
Root discriminant $2254.83$
Ramified primes $3, 17, 307$
Class number $11609505792$ (GRH)
Class group $[2, 2, 6, 6, 12, 12, 12, 36, 36, 36]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1418640313495891968, 0, 0, 0, 0, 0, 25321119032448, 0, 0, 0, 0, 0, 27592275, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 27592275*x^12 + 25321119032448*x^6 + 1418640313495891968)
 
gp: K = bnfinit(x^18 + 27592275*x^12 + 25321119032448*x^6 + 1418640313495891968, 1)
 

Normalized defining polynomial

\( x^{18} + 27592275 x^{12} + 25321119032448 x^{6} + 1418640313495891968 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2270120800429981380726586184495439361286008707872399584386003=-\,3^{33}\cdot 17^{12}\cdot 307^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2254.83$
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:  $3, 17, 307$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{17} a^{3}$, $\frac{1}{34} a^{4} - \frac{1}{2} a$, $\frac{1}{34} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{1734} a^{6} - \frac{1}{34} a^{3}$, $\frac{1}{5202} a^{7} + \frac{1}{6} a$, $\frac{1}{31212} a^{8} + \frac{1}{36} a^{2}$, $\frac{1}{6367248} a^{9} + \frac{73}{7344} a^{3} - \frac{1}{2}$, $\frac{1}{38203488} a^{10} - \frac{1}{10404} a^{7} - \frac{575}{44064} a^{4} + \frac{5}{12} a$, $\frac{1}{229220928} a^{11} - \frac{1}{62424} a^{8} + \frac{721}{264384} a^{5} - \frac{1}{72} a^{2}$, $\frac{1}{2131182494963712} a^{12} - \frac{104151023}{2458111297536} a^{6} - \frac{1}{34} a^{3} - \frac{18791}{60768}$, $\frac{1}{12787094969782272} a^{13} + \frac{1313444881}{14748667785216} a^{7} + \frac{41977}{364608} a$, $\frac{1}{76722569818693632} a^{14} + \frac{1313444881}{88492006711296} a^{8} + \frac{406585}{2187648} a^{2}$, $\frac{1}{15651404243013500928} a^{15} - \frac{1}{4262364989927424} a^{12} + \frac{1313444881}{18052369369104384} a^{9} - \frac{1313444881}{4916222595072} a^{6} + \frac{406585}{446280192} a^{3} - \frac{41977}{121536}$, $\frac{1}{93908425458081005568} a^{16} - \frac{1}{25574189939564544} a^{13} + \frac{1313444881}{108314216214626304} a^{10} - \frac{1313444881}{29497335570432} a^{7} + \frac{26658361}{2677681152} a^{4} - \frac{41977}{729216} a$, $\frac{1}{563450552748486033408} a^{17} - \frac{1}{153445139637387264} a^{14} + \frac{1313444881}{649885297287757824} a^{11} - \frac{1313444881}{176984013422592} a^{8} + \frac{105413689}{16066086912} a^{5} - \frac{406585}{4375296} a^{2}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{6}\times C_{6}\times C_{12}\times C_{12}\times C_{12}\times C_{36}\times C_{36}\times C_{36}$, which has order $11609505792$ (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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{199}{978212765188343808} a^{15} + \frac{6286519}{1128273085569024} a^{9} + \frac{107965}{27892512} a^{3} + \frac{1}{2} \) (order $6$)
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:  \( 392363527634279.25 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.6618824523.1 x3, 3.3.94249.1, 6.0.131426514198798532587.1, 6.0.239837598027.5, 6.0.1394460569330163.1 x2, Deg 9 x3

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

Sibling fields

Degree 6 sibling: data not computed
Degree 9 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.6.11.9$x^{6} + 3$$6$$1$$11$$S_3$$[5/2]_{2}$
3.6.11.9$x^{6} + 3$$6$$1$$11$$S_3$$[5/2]_{2}$
3.6.11.9$x^{6} + 3$$6$$1$$11$$S_3$$[5/2]_{2}$
$17$17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
307Data not computed