Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 + 7*x^4 - 8*x^3 + 43*x^2 - 42*x + 49)
gp: K = bnfinit(x^6 - x^5 + 7*x^4 - 8*x^3 + 43*x^2 - 42*x + 49, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49, -42, 43, -8, 7, -1, 1]);
\( x^{6} - x^{5} + 7x^{4} - 8x^{3} + 43x^{2} - 42x + 49 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
Invariants
| Degree: | $6$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[0, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: |
\(-3518667\)
\(\medspace = -\,3^{3}\cdot 19^{4}\)
| sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | \(12.33\) | 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: |
\(3\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $\card{ \Gal(K/\Q) }$: | $6$ | ||
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(57=3\cdot 19\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{57}(1,·)$, $\chi_{57}(20,·)$, $\chi_{57}(49,·)$, $\chi_{57}(7,·)$, $\chi_{57}(26,·)$, $\chi_{57}(11,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{7}a^{4}-\frac{1}{7}a$, $\frac{1}{259}a^{5}-\frac{1}{37}a^{4}+\frac{7}{37}a^{3}-\frac{43}{259}a^{2}+\frac{6}{37}a-\frac{5}{37}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
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: |
\( \frac{6}{259} a^{5} - \frac{5}{259} a^{4} + \frac{5}{37} a^{3} + \frac{1}{259} a^{2} + \frac{215}{259} a + \frac{7}{37} \)
(order $6$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
| Fundamental units: |
$\frac{30}{259}a^{5}-\frac{25}{259}a^{4}+\frac{25}{37}a^{3}-\frac{254}{259}a^{2}+\frac{1075}{259}a-\frac{150}{37}$, $\frac{5}{259}a^{5}-\frac{5}{37}a^{4}-\frac{2}{37}a^{3}-\frac{215}{259}a^{2}+\frac{30}{37}a-\frac{99}{37}$
| sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
| Regulator: | \( 7.80862678603 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: GaloisGroup(K);
| A cyclic group of order 6 |
| The 6 conjugacy class representatives for $C_6$ |
| Character table for $C_6$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 3.3.361.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | 3.3.361.1 $\times$ \(\Q(\sqrt{-3}) \) $\times$ \(\Q\) |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.6.0.1}{6} }$ | R | ${\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.1.0.1}{1} }^{6}$ | ${\href{/padicField/11.2.0.1}{2} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }$ | R | ${\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.1.0.1}{1} }^{6}$ | ${\href{/padicField/37.1.0.1}{1} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.6.0.1}{6} }$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.6.3.2 | $x^{6} - 9 x^{2} + 27$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
|
\(19\)
| 19.3.2.2 | $x^{3} - 19$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
| 19.3.2.2 | $x^{3} - 19$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.19.3t1.a.a | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 1.57.6t1.a.a | $1$ | $ 3 \cdot 19 $ | 6.0.3518667.1 | $C_6$ (as 6T1) | $0$ | $-1$ |
| * | 1.19.3t1.a.b | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 1.57.6t1.a.b | $1$ | $ 3 \cdot 19 $ | 6.0.3518667.1 | $C_6$ (as 6T1) | $0$ | $-1$ |
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 *.