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Properties

Label 9.3.470840338912...5792.1
Degree $9$
Signature $[3, 3]$
Discriminant $-\,2^{6}\cdot 7^{6}\cdot 673^{7}$
Root discriminant $919.71$
Ramified primes $2, 7, 673$
Class number $11340$ (GRH)
Class group $[3, 6, 630]$ (GRH)
Galois group $S_3\times C_3$ (as 9T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-648586550912, -111471767840, -8052829536, -328849540, -2242508, 388461, -6664, -1554, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 4*x^8 - 1554*x^7 - 6664*x^6 + 388461*x^5 - 2242508*x^4 - 328849540*x^3 - 8052829536*x^2 - 111471767840*x - 648586550912)
 
gp: K = bnfinit(x^9 - 4*x^8 - 1554*x^7 - 6664*x^6 + 388461*x^5 - 2242508*x^4 - 328849540*x^3 - 8052829536*x^2 - 111471767840*x - 648586550912, 1)
 

Normalized defining polynomial

\( x^{9} - 4 x^{8} - 1554 x^{7} - 6664 x^{6} + 388461 x^{5} - 2242508 x^{4} - 328849540 x^{3} - 8052829536 x^{2} - 111471767840 x - 648586550912 \)

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

Invariants

Degree:  $9$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-470840338912293629780425792=-\,2^{6}\cdot 7^{6}\cdot 673^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $919.71$
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:  $2, 7, 673$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{5} + \frac{1}{16} a^{3} - \frac{1}{8} a^{2}$, $\frac{1}{16} a^{6} + \frac{1}{16} a^{4} - \frac{1}{8} a^{3}$, $\frac{1}{608} a^{7} + \frac{1}{608} a^{6} - \frac{17}{608} a^{5} - \frac{49}{608} a^{4} - \frac{11}{152} a^{3} - \frac{25}{152} a^{2} - \frac{15}{38} a - \frac{7}{19}$, $\frac{1}{183986050569119234587364270846848} a^{8} + \frac{108335578584633033127282863025}{183986050569119234587364270846848} a^{7} + \frac{127279459476531147601993461243}{183986050569119234587364270846848} a^{6} - \frac{622991564892193364713048998137}{183986050569119234587364270846848} a^{5} - \frac{1086721964691027533957473464255}{22998256321139904323420533855856} a^{4} + \frac{4083479855808210174777447863633}{45996512642279808646841067711712} a^{3} + \frac{796943512376604609664558036511}{5749564080284976080855133463964} a^{2} - \frac{1417533758775007518123497442767}{5749564080284976080855133463964} a + \frac{601119758586011774123036972456}{1437391020071244020213783365991}$

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

Class group and class number

$C_{3}\times C_{6}\times C_{630}$, which has order $11340$ (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:  $5$
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:  \( 40152340.47454029 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 9T4):

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

3.3.22193521.2, 3.1.2692.1

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

Sibling fields

Galois closure: data not computed
Degree 6 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 ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{3}$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
673Data not computed