Properties

Label 6.2.69987605.1
Degree $6$
Signature $[2, 2]$
Discriminant $69987605$
Root discriminant $20.30$
Ramified primes $5, 241$
Class number $1$
Class group trivial
Galois group $C_3^2:D_4$ (as 6T13)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 2*x^5 + x^4 - 15*x^3 + 15*x^2 - 4)
 
gp: K = bnfinit(x^6 - 2*x^5 + x^4 - 15*x^3 + 15*x^2 - 4, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-4, 0, 15, -15, 1, -2, 1]);
 

\( x^{6} - 2 x^{5} + x^{4} - 15 x^{3} + 15 x^{2} - 4 \)

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

Invariants

Degree:  $6$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[2, 2]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(69987605\)\(\medspace = 5\cdot 241^{3}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $20.30$
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:  $5, 241$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{32} a^{5} + \frac{1}{8} a^{4} - \frac{7}{32} a^{3} + \frac{7}{32} a^{2} - \frac{7}{32} a - \frac{5}{16}$

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:  $3$
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:  \( \frac{1}{32} a^{5} + \frac{1}{8} a^{4} - \frac{7}{32} a^{3} - \frac{25}{32} a^{2} - \frac{39}{32} a + \frac{11}{16} \),  \( \frac{19}{32} a^{5} - \frac{5}{8} a^{4} - \frac{5}{32} a^{3} - \frac{283}{32} a^{2} + \frac{27}{32} a + \frac{33}{16} \),  \( \frac{2579}{32} a^{5} - \frac{229}{8} a^{4} - \frac{1573}{32} a^{3} - \frac{39451}{32} a^{2} - \frac{25381}{32} a - \frac{687}{16} \)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 144.165723986 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{2}\cdot 144.165723986 \cdot 1}{2\sqrt{69987605}}\approx 1.36063357574$

Galois group

$S_3\wr C_2$ (as 6T13):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 72
The 9 conjugacy class representatives for $C_3^2:D_4$
Character table for $C_3^2:D_4$

Intermediate fields

\(\Q(\sqrt{241}) \)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling algebras

Twin sextic algebra: 6.2.30125.1
Degree 6 sibling: 6.2.30125.1
Degree 9 sibling: 9.1.1749690125.1
Degree 12 siblings: 12.0.218711265625.1, Deg 12, Deg 12, Deg 12, Deg 12, Deg 12
Degree 18 siblings: Deg 18, Deg 18, Deg 18
Degree 24 siblings: data not computed
Degree 36 siblings: 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 ${\href{/LocalNumberField/2.3.0.1}{3} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.3.0.1}{3} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
241Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.1205.2t1.a.a$1$ $ 5 \cdot 241 $ $x^{2} - x - 301$ $C_2$ (as 2T1) $1$ $1$
* 1.241.2t1.a.a$1$ $ 241 $ $x^{2} - x - 60$ $C_2$ (as 2T1) $1$ $1$
1.5.2t1.a.a$1$ $ 5 $ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
2.1205.4t3.b.a$2$ $ 5 \cdot 241 $ $x^{4} - x^{3} + 8 x^{2} - 7 x + 19$ $D_{4}$ (as 4T3) $1$ $-2$
* 4.290405.6t13.b.a$4$ $ 5 \cdot 241^{2}$ $x^{6} - 2 x^{5} + x^{4} - 15 x^{3} + 15 x^{2} - 4$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.7260125.12t34.b.a$4$ $ 5^{3} \cdot 241^{2}$ $x^{6} - 2 x^{5} + x^{4} - 15 x^{3} + 15 x^{2} - 4$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.6025.6t13.b.a$4$ $ 5^{2} \cdot 241 $ $x^{6} - 2 x^{5} + x^{4} - 15 x^{3} + 15 x^{2} - 4$ $C_3^2:D_4$ (as 6T13) $1$ $0$
4.349938025.12t34.b.a$4$ $ 5^{2} \cdot 241^{3}$ $x^{6} - 2 x^{5} + x^{4} - 15 x^{3} + 15 x^{2} - 4$ $C_3^2:D_4$ (as 6T13) $1$ $0$

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 *.