magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1144, -868, 189, 89, -50, 10, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - 3*x^6 + 10*x^5 - 50*x^4 + 89*x^3 + 189*x^2 - 868*x + 1144)
gp: K = bnfinit(x^7 - 3*x^6 + 10*x^5 - 50*x^4 + 89*x^3 + 189*x^2 - 868*x + 1144, 1)
Normalized defining polynomial
\( x^{7} - 3 x^{6} + 10 x^{5} - 50 x^{4} + 89 x^{3} + 189 x^{2} - 868 x + 1144 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $7$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[1, 3]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-3372300719936=-\,2^{6}\cdot 23^{3}\cdot 163^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $61.62$ | 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: | $2, 23, 163$ | 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 \(a\))
$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{8} a^{2}$, $\frac{1}{64} a^{5} + \frac{1}{64} a^{4} - \frac{5}{64} a^{3} + \frac{7}{64} a^{2} - \frac{3}{16} a - \frac{3}{8}$, $\frac{1}{256} a^{6} - \frac{3}{128} a^{4} + \frac{11}{64} a^{3} + \frac{45}{256} a^{2} + \frac{21}{64} a + \frac{11}{32}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{29}$, which has order $29$
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: | \( \frac{1}{64} a^{6} - \frac{3}{32} a^{5} + \frac{5}{16} a^{4} - \frac{11}{32} a^{3} - \frac{93}{64} a^{2} + \frac{79}{16} a - \frac{27}{8} \), \( \frac{7}{64} a^{5} - \frac{25}{64} a^{4} + \frac{93}{64} a^{3} - \frac{495}{64} a^{2} + \frac{299}{16} a - \frac{101}{8} \), \( \frac{61}{256} a^{6} - \frac{81}{64} a^{5} + \frac{679}{128} a^{4} - \frac{387}{16} a^{3} + \frac{19805}{256} a^{2} - \frac{8723}{64} a + \frac{3659}{32} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 1224.29784805 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 14 |
| The 5 conjugacy class representatives for $D_{7}$ |
| Character table for $D_{7}$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.7.0.1}{7} }$ | ${\href{/LocalNumberField/5.7.0.1}{7} }$ | ${\href{/LocalNumberField/7.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ | ${\href{/LocalNumberField/17.7.0.1}{7} }$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ | R | ${\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | ${\href{/LocalNumberField/31.7.0.1}{7} }$ | ${\href{/LocalNumberField/37.7.0.1}{7} }$ | ${\href{/LocalNumberField/41.7.0.1}{7} }$ | ${\href{/LocalNumberField/43.7.0.1}{7} }$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ | ${\href{/LocalNumberField/59.7.0.1}{7} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 2.2.2.2 | $x^{2} + 2 x - 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.2 | $x^{2} + 2 x - 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.2 | $x^{2} + 2 x - 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| $23$ | $\Q_{23}$ | $x + 2$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 23.2.1.2 | $x^{2} + 46$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 46$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 23.2.1.2 | $x^{2} + 46$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| $163$ | $\Q_{163}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 163.2.1.1 | $x^{2} - 163$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 163.2.1.1 | $x^{2} - 163$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 163.2.1.1 | $x^{2} - 163$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| 1.2e2_23_163.2t1.1c1 | $1$ | $ 2^{2} \cdot 23 \cdot 163 $ | $x^{2} + 3749$ | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 2.2e2_23_163.7t2.1c1 | $2$ | $ 2^{2} \cdot 23 \cdot 163 $ | $x^{7} - 3 x^{6} + 10 x^{5} - 50 x^{4} + 89 x^{3} + 189 x^{2} - 868 x + 1144$ | $D_{7}$ (as 7T2) | $1$ | $0$ |
| * | 2.2e2_23_163.7t2.1c2 | $2$ | $ 2^{2} \cdot 23 \cdot 163 $ | $x^{7} - 3 x^{6} + 10 x^{5} - 50 x^{4} + 89 x^{3} + 189 x^{2} - 868 x + 1144$ | $D_{7}$ (as 7T2) | $1$ | $0$ |
| * | 2.2e2_23_163.7t2.1c3 | $2$ | $ 2^{2} \cdot 23 \cdot 163 $ | $x^{7} - 3 x^{6} + 10 x^{5} - 50 x^{4} + 89 x^{3} + 189 x^{2} - 868 x + 1144$ | $D_{7}$ (as 7T2) | $1$ | $0$ |
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 *.