↑

Properties

Label 6.2.2451110928.1
Degree $6$
Signature $[2, 2]$
Discriminant $2^{4}\cdot 3\cdot 7^{3}\cdot 53^{3}$
Root discriminant $36.72$
Ramified primes $2, 3, 7, 53$
Class number $2$
Class group $[2]$
Galois group $S_4\times C_2$ (as 6T11)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-378, 0, 72, 5, -11, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - 3*x^5 - 11*x^4 + 5*x^3 + 72*x^2 - 378)
 
gp: K = bnfinit(x^6 - 3*x^5 - 11*x^4 + 5*x^3 + 72*x^2 - 378, 1)
 

Normalized defining polynomial

\( x^{6} - 3 x^{5} - 11 x^{4} + 5 x^{3} + 72 x^{2} - 378 \)

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:  \(2451110928=2^{4}\cdot 3\cdot 7^{3}\cdot 53^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.72$
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:  $2, 3, 7, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3699} a^{5} - \frac{18}{137} a^{4} + \frac{1690}{3699} a^{3} + \frac{1214}{3699} a^{2} - \frac{616}{1233} a + \frac{125}{411}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{194}{1233} a^{5} + \frac{82}{411} a^{4} - \frac{118}{1233} a^{3} - \frac{2864}{1233} a^{2} + \frac{78}{137} a + \frac{2467}{137} \),  \( \frac{3478}{3699} a^{5} - \frac{2177}{411} a^{4} + \frac{7507}{3699} a^{3} + \frac{7898}{3699} a^{2} + \frac{81062}{1233} a - \frac{68725}{411} \),  \( \frac{21}{137} a^{5} + \frac{344}{411} a^{4} - \frac{130}{137} a^{3} - \frac{1060}{411} a^{2} - \frac{1481}{411} a + \frac{4993}{137} \)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1038.52486022 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_4$ (as 6T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 48
The 10 conjugacy class representatives for $S_4\times C_2$
Character table for $S_4\times C_2$

Intermediate fields

3.1.1484.1

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

Sibling algebras

Twin sextic algebra: 4.2.13356.1 $\times$ \(\Q(\sqrt{-3}) \)
Degree 6 sibling: 6.0.6606768.1
Degree 8 siblings: 8.0.178382736.1, 8.4.24552778165776.4
Degree 12 siblings: Deg 12, Deg 12, Deg 12, Deg 12, Deg 12, Deg 12
Degree 16 sibling: Deg 16
Degree 24 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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
53Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.3.2t1.1c1$1$ $ 3 $ $x^{2} - x + 1$ $C_2$ (as 2T1) $1$ $-1$
1.7_53.2t1.1c1$1$ $ 7 \cdot 53 $ $x^{2} - x + 93$ $C_2$ (as 2T1) $1$ $-1$
1.3_7_53.2t1.1c1$1$ $ 3 \cdot 7 \cdot 53 $ $x^{2} - x - 278$ $C_2$ (as 2T1) $1$ $1$
2.2e2_3e2_7_53.6t3.2c1$2$ $ 2^{2} \cdot 3^{2} \cdot 7 \cdot 53 $ $x^{6} - 8 x^{4} - 24 x^{3} + 64 x^{2} + 96 x + 144$ $D_{6}$ (as 6T3) $1$ $0$
* 2.2e2_7_53.3t2.1c1$2$ $ 2^{2} \cdot 7 \cdot 53 $ $x^{3} + 8 x - 12$ $S_3$ (as 3T2) $1$ $0$
3.2e2_3e2_7_53.4t5.1c1$3$ $ 2^{2} \cdot 3^{2} \cdot 7 \cdot 53 $ $x^{4} - x^{3} - 7 x^{2} + 7 x - 2$ $S_4$ (as 4T5) $1$ $1$
* 3.2e2_3_7e2_53e2.6t11.1c1$3$ $ 2^{2} \cdot 3 \cdot 7^{2} \cdot 53^{2}$ $x^{6} - 3 x^{5} - 11 x^{4} + 5 x^{3} + 72 x^{2} - 378$ $S_4\times C_2$ (as 6T11) $1$ $1$
3.2e2_3_7_53.6t11.1c1$3$ $ 2^{2} \cdot 3 \cdot 7 \cdot 53 $ $x^{6} - 3 x^{5} - 11 x^{4} + 5 x^{3} + 72 x^{2} - 378$ $S_4\times C_2$ (as 6T11) $1$ $-1$
3.2e2_3e2_7e2_53e2.6t8.1c1$3$ $ 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 53^{2}$ $x^{4} - x^{3} - 7 x^{2} + 7 x - 2$ $S_4$ (as 4T5) $1$ $-1$

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 *.