# Properties

 Label 3.3.2700.1 Degree $3$ Signature $[3, 0]$ Discriminant $2700$ Root discriminant $13.92$ Ramified primes $2, 3, 5$ Class number $1$ Class group trivial Galois group $S_3$ (as 3T2)

# Related objects

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^3 - 15*x - 20)

gp: K = bnfinit(x^3 - 15*x - 20, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-20, -15, 0, 1]);

$$x^{3} - 15 x - 20$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $3$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[3, 0]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$2700$$$$\medspace = 2^{2}\cdot 3^{3}\cdot 5^{2}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $13.92$ 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: $2, 3, 5$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $1$ This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $a^{2}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## 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: $2$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-1$$ (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $$a^{2} + a - 1$$,  $$8 a^{2} - 22 a - 59$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$20.369355187$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{3}\cdot(2\pi)^{0}\cdot 20.369355187 \cdot 1}{2\sqrt{2700}}\approx 1.5680336934$

## 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: 6.6.87480000.1

## Multiplicative Galois module structure

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

## 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 R ${\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.3.0.1}{3} }$ ${\href{/padicField/13.3.0.1}{3} }$ ${\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.3.0.1}{3} }$ ${\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.3.0.1}{3} }$ ${\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.3.0.1}{3} }$ ${\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.3.0.1}{3} }$

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$$\Q_{2}$$x + 1$$1$$1$$0Trivial[\ ] 2.2.2.2x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
$3$3.3.3.1$x^{3} + 6 x + 3$$3$$1$$3$$S_3$$[3/2]_{2} 55.3.2.1x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
1.12.2t1.a.a$1$ $2^{2} \cdot 3$ $$\Q(\sqrt{3})$$ $C_2$ (as 2T1) $1$ $1$
* 2.2700.3t2.a.a$2$ $2^{2} \cdot 3^{3} \cdot 5^{2}$ 3.3.2700.1 $S_3$ (as 3T2) $1$ $2$

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 *.