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Properties

Label 18.0.79593402605...7707.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{31}\cdot 7^{12}\cdot 13^{12}\cdot 43^{12}$
Root discriminant $1647.08$
Ramified primes $3, 7, 13, 43$
Class number $11249543088$ (GRH)
Class group $[3, 3, 3, 9, 9, 18, 18, 126, 126]$ (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![1855425871872, 0, 0, 0, 0, 0, 3825681705, 0, 0, 0, 0, 0, 248166, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 248166*x^12 + 3825681705*x^6 + 1855425871872)
 
gp: K = bnfinit(x^18 + 248166*x^12 + 3825681705*x^6 + 1855425871872, 1)
 

Normalized defining polynomial

\( x^{18} + 248166 x^{12} + 3825681705 x^{6} + 1855425871872 \)

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:  \(-7959340260508580929689411840109697873368492825644925717707=-\,3^{31}\cdot 7^{12}\cdot 13^{12}\cdot 43^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1647.08$
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:  $3, 7, 13, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{18} a^{6} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{576} a^{7} + \frac{1}{6} a^{5} - \frac{1}{2} a^{2} - \frac{47}{192} a$, $\frac{1}{1152} a^{8} - \frac{47}{384} a^{2}$, $\frac{1}{1152} a^{9} - \frac{47}{384} a^{3}$, $\frac{1}{1152} a^{10} - \frac{47}{384} a^{4}$, $\frac{1}{2304} a^{11} - \frac{1}{2304} a^{8} - \frac{47}{768} a^{5} + \frac{47}{768} a^{2}$, $\frac{1}{43827754752} a^{12} + \frac{1}{3456} a^{10} - \frac{1}{2304} a^{9} + \frac{219291857}{14609251584} a^{6} - \frac{239}{1152} a^{4} + \frac{47}{768} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a + \frac{2281414}{19022463}$, $\frac{1}{43827754752} a^{13} - \frac{1}{6912} a^{11} - \frac{1}{2304} a^{10} - \frac{1}{2304} a^{8} - \frac{8977699}{14609251584} a^{7} - \frac{337}{2304} a^{5} + \frac{47}{768} a^{4} + \frac{1}{3} a^{3} - \frac{337}{768} a^{2} + \frac{393302515}{1217437632} a$, $\frac{1}{701244076032} a^{14} + \frac{49408129}{116874012672} a^{8} + \frac{1}{6} a^{4} + \frac{30595147777}{77916008448} a^{2} - \frac{1}{2} a$, $\frac{1}{89759241732096} a^{15} - \frac{5124701807}{14959873622016} a^{9} - \frac{1}{6} a^{5} + \frac{4023181861921}{9973249081344} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{5744591470854144} a^{16} - \frac{57068707439}{957431911809024} a^{10} - \frac{1}{1152} a^{7} + \frac{1}{6} a^{5} - \frac{26000453393375}{638287941206016} a^{4} - \frac{1}{2} a^{2} + \frac{47}{384} a$, $\frac{1}{1102961562403995648} a^{17} - \frac{1}{1402488152064} a^{14} + \frac{6591764013457}{183826927067332608} a^{11} + \frac{1}{3456} a^{9} - \frac{49408129}{233748025344} a^{8} - \frac{26182508377398239}{122551284711555072} a^{5} + \frac{1}{6} a^{4} + \frac{337}{1152} a^{3} + \frac{47320860671}{155832016896} a^{2} - \frac{1}{6} a$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{9}\times C_{9}\times C_{18}\times C_{18}\times C_{126}\times C_{126}$, which has order $11249543088$ (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:  \( -\frac{1}{114635046912} a^{15} - \frac{126131}{57317523456} a^{9} - \frac{1520139363}{38211682304} a^{3} + \frac{1}{2} \) (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:  \( 10228947091425.262 \) (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

\(\Q(\sqrt{-3}) \), 3.1.413412363.1 x3, Deg 3, 6.0.512729345643731307.1, 6.0.4614564110793581763.3, 6.0.301377612627.3 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
3Data not computed
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$13$13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$43$43.9.6.1$x^{9} + 1290 x^{6} + 552851 x^{3} + 79507000$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
43.9.6.1$x^{9} + 1290 x^{6} + 552851 x^{3} + 79507000$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$