Properties

Label 6.0.84015792.1
Degree $6$
Signature $[0, 3]$
Discriminant $-84015792$
Root discriminant $20.93$
Ramified primes $2, 3, 7$
Class number $9$
Class group $[3, 3]$
Galois group $S_3$ (as 6T2)

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Show commands: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 3*x^5 + 6*x^4 + 21*x^3 - 36*x^2 - 45*x + 225)
 
gp: K = bnfinit(x^6 - 3*x^5 + 6*x^4 + 21*x^3 - 36*x^2 - 45*x + 225, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![225, -45, -36, 21, 6, -3, 1]);
 

\(x^{6} - 3 x^{5} + 6 x^{4} + 21 x^{3} - 36 x^{2} - 45 x + 225\)  Toggle raw display

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

Invariants

Degree:  $6$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 3]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-84015792\)\(\medspace = -\,2^{4}\cdot 3^{7}\cdot 7^{4}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $20.93$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 3, 7$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $6$
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{585} a^{5} - \frac{28}{585} a^{4} - \frac{1}{65} a^{3} + \frac{82}{195} a^{2} - \frac{47}{195} a - \frac{5}{13}$  Toggle raw display

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $2$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{2}{585} a^{5} - \frac{1}{65} a^{4} + \frac{2}{65} a^{3} - \frac{34}{195} a^{2} - \frac{12}{65} a + \frac{10}{13} \) (order $6$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  \( \frac{2}{585} a^{5} - \frac{56}{585} a^{4} - \frac{2}{65} a^{3} - \frac{31}{195} a^{2} - \frac{484}{195} a - \frac{62}{13} \),  \( \frac{1}{65} a^{5} + \frac{8}{585} a^{4} - \frac{9}{65} a^{3} + \frac{23}{195} a^{2} + \frac{97}{195} a - \frac{6}{13} \)  Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 19.940806802 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{3}\cdot 19.940806802 \cdot 9}{6\sqrt{84015792}}\approx 0.80945577728$

Galois group

$S_3$ (as 6T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 6
The 3 conjugacy class representatives for $S_3$
Character table for $S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.5292.1 x3

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

Sibling algebras

Twin sextic algebra: 3.1.5292.1 $\times$ \(\Q\) $\times$ \(\Q\) $\times$ \(\Q\)
Degree 3 sibling: 3.1.5292.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.2.0.1}{2} }^{3}$ R ${\href{/padicField/11.2.0.1}{2} }^{3}$ ${\href{/padicField/13.1.0.1}{1} }^{6}$ ${\href{/padicField/17.2.0.1}{2} }^{3}$ ${\href{/padicField/19.3.0.1}{3} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{3}$ ${\href{/padicField/29.2.0.1}{2} }^{3}$ ${\href{/padicField/31.3.0.1}{3} }^{2}$ ${\href{/padicField/37.1.0.1}{1} }^{6}$ ${\href{/padicField/41.2.0.1}{2} }^{3}$ ${\href{/padicField/43.3.0.1}{3} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{3}$ ${\href{/padicField/53.2.0.1}{2} }^{3}$ ${\href{/padicField/59.2.0.1}{2} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$3$3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
$7$7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$