# Properties

 Label 6.6.192293689.1 Degree $6$ Signature $[6, 0]$ Discriminant $7^{4}\cdot 283^{2}$ Root discriminant $24.02$ Ramified primes $7, 283$ Class number $1$ Class group Trivial Galois group $A_6$ (as 6T15)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 - 12*x^4 + 7*x^3 + 21*x^2 - 7*x - 7)

gp: K = bnfinit(x^6 - x^5 - 12*x^4 + 7*x^3 + 21*x^2 - 7*x - 7, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-7, -7, 21, 7, -12, -1, 1]);

## Normalizeddefining polynomial

$$x^{6} - x^{5} - 12 x^{4} + 7 x^{3} + 21 x^{2} - 7 x - 7$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $6$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[6, 0]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$192293689=7^{4}\cdot 283^{2}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $24.02$ 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: $7, 283$ 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 $$a$$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{17} a^{5} + \frac{4}{17} a^{4} + \frac{8}{17} a^{3} - \frac{4}{17} a^{2} + \frac{1}{17} a - \frac{2}{17}$

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: $5$ 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: Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$279.295287926$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Galois group

$A_6$ (as 6T15):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A non-solvable group of order 360 The 7 conjugacy class representatives for $A_6$ Character table for $A_6$

## Intermediate fields

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

## Sibling algebras

 Twin sextic algebra: 6.6.15400609258321.1 Degree 6 sibling: 6.6.15400609258321.1 Degree 10 sibling: 10.10.60437550349593857881.1 Degree 15 siblings: 15.15.11621759510846642583679213009.1, Deg 15 Degree 20 sibling: Deg 20 Degree 30 siblings: data not computed Degree 36 sibling: data not computed Degree 40 sibling: data not computed Degree 45 sibling: data not computed

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 ${\href{/LocalNumberField/2.5.0.1}{5} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ R ${\href{/LocalNumberField/11.5.0.1}{5} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.5.0.1}{5} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2} 7.4.3.2x^{4} - 7$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
283Data not computed

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
5.15400609258321.6t15.a.a$5$ $7^{4} \cdot 283^{4}$ $x^{6} - x^{5} - 12 x^{4} + 7 x^{3} + 21 x^{2} - 7 x - 7$ $A_6$ (as 6T15) $1$ $5$
* 5.192293689.6t15.a.a$5$ $7^{4} \cdot 283^{2}$ $x^{6} - x^{5} - 12 x^{4} + 7 x^{3} + 21 x^{2} - 7 x - 7$ $A_6$ (as 6T15) $1$ $5$
8.60437550349593857881.36t555.a.a$8$ $7^{6} \cdot 283^{6}$ $x^{6} - x^{5} - 12 x^{4} + 7 x^{3} + 21 x^{2} - 7 x - 7$ $A_6$ (as 6T15) $1$ $8$
8.60437550349593857881.36t555.a.b$8$ $7^{6} \cdot 283^{6}$ $x^{6} - x^{5} - 12 x^{4} + 7 x^{3} + 21 x^{2} - 7 x - 7$ $A_6$ (as 6T15) $1$ $8$
9.60437550349593857881.10t26.a.a$9$ $7^{6} \cdot 283^{6}$ $x^{6} - x^{5} - 12 x^{4} + 7 x^{3} + 21 x^{2} - 7 x - 7$ $A_6$ (as 6T15) $1$ $9$
10.2961439967130099036169.30t88.a.a$10$ $7^{8} \cdot 283^{6}$ $x^{6} - x^{5} - 12 x^{4} + 7 x^{3} + 21 x^{2} - 7 x - 7$ $A_6$ (as 6T15) $1$ $10$

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 *.