Properties

Label 5.1.2640625.1
Degree $5$
Signature $[1, 2]$
Discriminant $2640625$
Root discriminant $19.25$
Ramified primes $5, 13$
Class number $3$
Class group $[3]$
Galois group $A_5$ (as 5T4)

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Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^5 + 5*x^3 - 10*x^2 - 45)
 
gp: K = bnfinit(x^5 + 5*x^3 - 10*x^2 - 45, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-45, 0, -10, 5, 0, 1]);
 

\( x^{5} + 5 x^{3} - 10 x^{2} - 45 \)

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

Invariants

Degree:  $5$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 2]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(2640625\)\(\medspace = 5^{6}\cdot 13^{2}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $19.25$
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:  $5, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{21} a^{4} - \frac{2}{7} a^{3} - \frac{1}{21} a^{2} - \frac{4}{21} a + \frac{1}{7}$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( \frac{4}{21} a^{4} - \frac{1}{7} a^{3} + \frac{17}{21} a^{2} - \frac{16}{21} a - \frac{3}{7} \),  \( \frac{1}{7} a^{4} + \frac{1}{7} a^{3} - \frac{1}{7} a^{2} + \frac{3}{7} a - \frac{32}{7} \)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 12.2810502337 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{2}\cdot 12.2810502337 \cdot 3}{2\sqrt{2640625}}\approx 0.895082639531$

Galois group

$A_5$ (as 5T4):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 60
The 5 conjugacy class representatives for $A_5$
Character table for $A_5$

Intermediate fields

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

Sibling fields

Degree 6 sibling: 6.2.2640625.1
Degree 10 sibling: 10.2.6972900390625.1
Degree 12 sibling: Deg 12
Degree 15 sibling: Deg 15
Degree 20 sibling: Deg 20
Degree 30 sibling: Deg 30

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.5.0.1}{5} }$ ${\href{/LocalNumberField/3.3.0.1}{3} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }$ R ${\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }$ ${\href{/LocalNumberField/31.5.0.1}{5} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }$ ${\href{/LocalNumberField/47.5.0.1}{5} }$ ${\href{/LocalNumberField/53.5.0.1}{5} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.5.6.2$x^{5} + 15 x^{2} + 5$$5$$1$$6$$D_{5}$$[3/2]_{2}$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
3.105625.12t33.a.a$3$ $ 5^{4} \cdot 13^{2}$ $x^{5} + 5 x^{3} - 10 x^{2} - 45$ $A_5$ (as 5T4) $1$ $-1$
3.105625.12t33.a.b$3$ $ 5^{4} \cdot 13^{2}$ $x^{5} + 5 x^{3} - 10 x^{2} - 45$ $A_5$ (as 5T4) $1$ $-1$
* 4.2640625.5t4.a.a$4$ $ 5^{6} \cdot 13^{2}$ $x^{5} + 5 x^{3} - 10 x^{2} - 45$ $A_5$ (as 5T4) $1$ $0$
5.2640625.6t12.a.a$5$ $ 5^{6} \cdot 13^{2}$ $x^{5} + 5 x^{3} - 10 x^{2} - 45$ $A_5$ (as 5T4) $1$ $1$

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 *.