Degree $3$
Signature $[1, 1]$
Discriminant $-243$
Root discriminant \(6.24\)
Ramified prime see page
Class number $1$
Class group trivial
Galois group $S_3$ (as 3T2)

Related objects


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Show commands: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^3 - 3)
gp: K = bnfinit(x^3 - 3, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 0, 0, 1]);

\( x^{3} - 3 \) Copy content Toggle raw display

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


Degree:  $3$
gp: poldegree(K.pol)
magma: Degree(K);
Signature:  $[1, 1]$
sage: K.signature()
gp: K.sign
magma: Signature(K);
Discriminant:   \(-243\) \(\medspace = -\,3^{5}\) Copy content Toggle raw display
sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
Root discriminant:  \(6.24\)
sage: (K.disc().abs())^(1./
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
Ramified primes:   \(3\) Copy content Toggle raw display
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
$\card{ \Aut(K/\Q) }$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$ Copy content Toggle raw display

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

Monogenic:  Yes
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$

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

Unit group

sage: UK = K.unit_group()
magma: UK, f := UnitGroup(K);
Rank:  $1$
sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
Fundamental unit:   $a^{2}-2$ Copy content Toggle raw display
sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
Regulator:  \( 2.52468140471 \)
sage: K.regulator()
gp: K.reg
magma: Regulator(K);

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{1}\cdot 2.52468140471 \cdot 1}{2\sqrt{243}}\approx 1.01761456150$

Galois group

$S_3$ (as 3T2):

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

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Galois closure: 6.0.177147.2

Multiplicative Galois module structure

$U_{K^{gal}}/\textrm{Tors}(U_{K^{gal}}) \cong$ $A'$

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/}{2} }{,}\,{\href{/padicField/}{1} }$ R ${\href{/padicField/}{2} }{,}\,{\href{/padicField/}{1} }$ ${\href{/padicField/}{3} }$ ${\href{/padicField/}{2} }{,}\,{\href{/padicField/}{1} }$ ${\href{/padicField/}{3} }$ ${\href{/padicField/}{2} }{,}\,{\href{/padicField/}{1} }$ ${\href{/padicField/}{3} }$ ${\href{/padicField/}{2} }{,}\,{\href{/padicField/}{1} }$ ${\href{/padicField/}{2} }{,}\,{\href{/padicField/}{1} }$ ${\href{/padicField/}{3} }$ ${\href{/padicField/}{3} }$ ${\href{/padicField/}{2} }{,}\,{\href{/padicField/}{1} }$ ${\href{/padicField/}{3} }$ ${\href{/padicField/}{2} }{,}\,{\href{/padicField/}{1} }$ ${\href{/padicField/}{2} }{,}\,{\href{/padicField/}{1} }$ ${\href{/padicField/}{2} }{,}\,{\href{/padicField/}{1} }$

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
\(3\) Copy content Toggle raw display$x^{3} + 3$$3$$1$$5$$S_3$$[5/2]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.243.3t2.b.a$2$ $ 3^{5}$ $S_3$ (as 3T2) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.