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Properties

Label 5.5.1330863361.1
Degree $5$
Signature $[5, 0]$
Discriminant $191^{4}$
Root discriminant $66.81$
Ramified prime $191$
Class number $11$
Class group $[11]$
Galois group $C_5$ (as 5T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![155, -437, 359, -76, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^5 - x^4 - 76*x^3 + 359*x^2 - 437*x + 155)
 
gp: K = bnfinit(x^5 - x^4 - 76*x^3 + 359*x^2 - 437*x + 155, 1)
 

Normalized defining polynomial

\( x^{5} - x^{4} - 76 x^{3} + 359 x^{2} - 437 x + 155 \)

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

Invariants

Degree:  $5$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[5, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1330863361=191^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.81$
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:  $191$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(191\)
Dirichlet character group:    $\lbrace$$\chi_{191}(184,·)$, $\chi_{191}(1,·)$, $\chi_{191}(39,·)$, $\chi_{191}(109,·)$, $\chi_{191}(49,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{55} a^{4} - \frac{14}{55} a^{3} - \frac{4}{55} a^{2} + \frac{26}{55} a - \frac{1}{11}$

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

Class group and class number

$C_{11}$, which has order $11$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $4$
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:  \( a - 1 \),  \( \frac{3}{55} a^{4} + \frac{13}{55} a^{3} - \frac{177}{55} a^{2} + \frac{78}{55} a + \frac{52}{11} \),  \( \frac{9}{55} a^{4} - \frac{16}{55} a^{3} - \frac{751}{55} a^{2} + \frac{3424}{55} a - \frac{526}{11} \),  \( \frac{16}{55} a^{4} - \frac{4}{55} a^{3} - \frac{1219}{55} a^{2} + \frac{4816}{55} a - \frac{665}{11} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 185.439033921 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5$ (as 5T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 5
The 5 conjugacy class representatives for $C_5$
Character table for $C_5$

Intermediate fields

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }$ ${\href{/LocalNumberField/3.5.0.1}{5} }$ ${\href{/LocalNumberField/5.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/7.5.0.1}{5} }$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/13.5.0.1}{5} }$ ${\href{/LocalNumberField/17.5.0.1}{5} }$ ${\href{/LocalNumberField/19.5.0.1}{5} }$ ${\href{/LocalNumberField/23.5.0.1}{5} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.5.0.1}{5} }$

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
$191$191.5.4.1$x^{5} - 191$$5$$1$$4$$C_5$$[\ ]_{5}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.191.5t1.1c1$1$ $ 191 $ $x^{5} - x^{4} - 76 x^{3} + 359 x^{2} - 437 x + 155$ $C_5$ (as 5T1) $0$ $1$
* 1.191.5t1.1c2$1$ $ 191 $ $x^{5} - x^{4} - 76 x^{3} + 359 x^{2} - 437 x + 155$ $C_5$ (as 5T1) $0$ $1$
* 1.191.5t1.1c3$1$ $ 191 $ $x^{5} - x^{4} - 76 x^{3} + 359 x^{2} - 437 x + 155$ $C_5$ (as 5T1) $0$ $1$
* 1.191.5t1.1c4$1$ $ 191 $ $x^{5} - x^{4} - 76 x^{3} + 359 x^{2} - 437 x + 155$ $C_5$ (as 5T1) $0$ $1$

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 *.