magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25, 5, 24, -2, 7, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^6 - x^5 + 7*x^4 - 2*x^3 + 24*x^2 + 5*x + 25)
gp: K = bnfinit(x^6 - x^5 + 7*x^4 - 2*x^3 + 24*x^2 + 5*x + 25, 1)
Normalized defining polynomial
\( x^{6} - x^{5} + 7 x^{4} - 2 x^{3} + 24 x^{2} + 5 x + 25 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $6$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 3]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-936859=-\,7\cdot 11\cdot 23^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $9.89$ | 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: | $7, 11, 23$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{125} a^{5} + \frac{14}{125} a^{4} - \frac{33}{125} a^{3} + \frac{3}{125} a^{2} - \frac{56}{125} a + \frac{8}{25}$
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: | $2$ | 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: | \( \frac{1}{125} a^{5} + \frac{14}{125} a^{4} - \frac{33}{125} a^{3} + \frac{128}{125} a^{2} - \frac{56}{125} a + \frac{58}{25} \), \( \frac{7}{125} a^{5} - \frac{27}{125} a^{4} + \frac{19}{125} a^{3} - \frac{104}{125} a^{2} - \frac{17}{125} a - \frac{19}{25} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 1.73686857052 \) | 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
| 3.1.23.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | 4.2.136367.1 $\times$ \(\Q(\sqrt{77}) \) |
| Degree 6 sibling: | 6.2.40733.1 |
| Degree 8 siblings: | 8.0.9837262146481.3, 8.4.18595958689.1 |
| 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 | ${\href{/LocalNumberField/2.6.0.1}{6} }$ | ${\href{/LocalNumberField/3.6.0.1}{6} }$ | ${\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ | R | R | ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | R | ${\href{/LocalNumberField/29.6.0.1}{6} }$ | ${\href{/LocalNumberField/31.6.0.1}{6} }$ | ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ | ${\href{/LocalNumberField/47.6.0.1}{6} }$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $7$ | 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.2.1.2 | $x^{2} + 14$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| $11$ | 11.2.1.2 | $x^{2} + 33$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 11.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| $23$ | 23.2.1.2 | $x^{2} + 46$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 23.4.2.1 | $x^{4} + 299 x^{2} + 25921$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| 1.7_11.2t1.1c1 | $1$ | $ 7 \cdot 11 $ | $x^{2} - x - 19$ | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.23.2t1.1c1 | $1$ | $ 23 $ | $x^{2} - x + 6$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.7_11_23.2t1.1c1 | $1$ | $ 7 \cdot 11 \cdot 23 $ | $x^{2} - x + 443$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 2.7e2_11e2_23.6t3.1c1 | $2$ | $ 7^{2} \cdot 11^{2} \cdot 23 $ | $x^{6} - x^{5} - 58 x^{4} + 37 x^{3} + 1103 x^{2} - 476 x - 6839$ | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| * | 2.23.3t2.1c1 | $2$ | $ 23 $ | $x^{3} - x^{2} + 1$ | $S_3$ (as 3T2) | $1$ | $0$ |
| 3.7e2_11e2_23.4t5.1c1 | $3$ | $ 7^{2} \cdot 11^{2} \cdot 23 $ | $x^{4} - x^{3} - 5 x^{2} - 7 x - 28$ | $S_4$ (as 4T5) | $1$ | $1$ | |
| * | 3.7_11_23e2.6t11.2c1 | $3$ | $ 7 \cdot 11 \cdot 23^{2}$ | $x^{6} - x^{5} + 7 x^{4} - 2 x^{3} + 24 x^{2} + 5 x + 25$ | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ |
| 3.7_11_23.6t11.2c1 | $3$ | $ 7 \cdot 11 \cdot 23 $ | $x^{6} - x^{5} + 7 x^{4} - 2 x^{3} + 24 x^{2} + 5 x + 25$ | $S_4\times C_2$ (as 6T11) | $1$ | $1$ | |
| 3.7e2_11e2_23e2.6t8.1c1 | $3$ | $ 7^{2} \cdot 11^{2} \cdot 23^{2}$ | $x^{4} - x^{3} - 5 x^{2} - 7 x - 28$ | $S_4$ (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 *.