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Properties

Label 6.2.100000000.1
Degree $6$
Signature $[2, 2]$
Discriminant $2^{8}\cdot 5^{8}$
Root discriminant $21.54$
Ramified primes $2, 5$
Class number $1$
Class group Trivial
Galois group $\PSL(2,5)$ (as 6T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7, 6, -45, 20, -5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 - 5*x^4 + 20*x^3 - 45*x^2 + 6*x - 7)
 
gp: K = bnfinit(x^6 - 2*x^5 - 5*x^4 + 20*x^3 - 45*x^2 + 6*x - 7, 1)
 

Normalized defining polynomial

\( x^{6} - 2 x^{5} - 5 x^{4} + 20 x^{3} - 45 x^{2} + 6 x - 7 \)

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

Invariants

Degree:  $6$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(100000000=2^{8}\cdot 5^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.54$
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:  $2, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{8} a^{4} - \frac{1}{2} a + \frac{3}{8}$, $\frac{1}{8} a^{5} + \frac{3}{8} a - \frac{1}{2}$

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $3$
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:  \( \frac{9}{8} a^{5} - \frac{13}{8} a^{4} - 7 a^{3} + \frac{39}{2} a^{2} - \frac{297}{8} a - \frac{191}{8} \),  \( \frac{1}{8} a^{5} - \frac{3}{8} a^{4} - a^{3} + \frac{5}{2} a^{2} - \frac{57}{8} a - \frac{25}{8} \),  \( \frac{3}{4} a^{3} - \frac{1}{4} a^{2} + \frac{3}{4} a + \frac{23}{4} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 342.859495562 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$A_5$ (as 6T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 60
The 5 conjugacy class representatives for $\PSL(2,5)$
Character table for $\PSL(2,5)$

Intermediate fields

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

Sibling algebras

Twin sextic algebra: 5.1.100000000.1 $\times$ \(\Q\)
Degree 5 sibling: 5.1.100000000.1
Degree 10 sibling: 10.2.10000000000000000.2
Degree 12 sibling: 12.0.2560000000000000000.23
Degree 15 sibling: Deg 15
Degree 20 sibling: Deg 20
Degree 30 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/7.5.0.1}{5} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

In the table, R denotes a ramified prime. 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
$2$2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.5.8.6$x^{5} + 5 x^{4} + 5$$5$$1$$8$$D_{5}$$[2]^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
3.2e8_5e4.12t33.2c1$3$ $ 2^{8} \cdot 5^{4}$ $x^{6} - 2 x^{5} - 5 x^{4} + 20 x^{3} - 45 x^{2} + 6 x - 7$ $\PSL(2,5)$ (as 6T12) $1$ $-1$
3.2e8_5e4.12t33.2c2$3$ $ 2^{8} \cdot 5^{4}$ $x^{6} - 2 x^{5} - 5 x^{4} + 20 x^{3} - 45 x^{2} + 6 x - 7$ $\PSL(2,5)$ (as 6T12) $1$ $-1$
4.2e8_5e8.5t4.2c1$4$ $ 2^{8} \cdot 5^{8}$ $x^{6} - 2 x^{5} - 5 x^{4} + 20 x^{3} - 45 x^{2} + 6 x - 7$ $\PSL(2,5)$ (as 6T12) $1$ $0$
* 5.2e8_5e8.6t12.2c1$5$ $ 2^{8} \cdot 5^{8}$ $x^{6} - 2 x^{5} - 5 x^{4} + 20 x^{3} - 45 x^{2} + 6 x - 7$ $\PSL(2,5)$ (as 6T12) $1$ $1$

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 *.