magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14, -56, 120, -144, 113, -52, 22, -4, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 - 4*x^7 + 22*x^6 - 52*x^5 + 113*x^4 - 144*x^3 + 120*x^2 - 56*x + 14)
gp: K = bnfinit(x^8 - 4*x^7 + 22*x^6 - 52*x^5 + 113*x^4 - 144*x^3 + 120*x^2 - 56*x + 14, 1)
Normalized defining polynomial
\( x^{8} - 4 x^{7} + 22 x^{6} - 52 x^{5} + 113 x^{4} - 144 x^{3} + 120 x^{2} - 56 x + 14 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $8$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(10070523904=2^{22}\cdot 7^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $17.80$ | 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: | $2, 7$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(112=2^{4}\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{112}(1,·)$, $\chi_{112}(99,·)$, $\chi_{112}(97,·)$, $\chi_{112}(41,·)$, $\chi_{112}(43,·)$, $\chi_{112}(83,·)$, $\chi_{112}(57,·)$, $\chi_{112}(27,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{31} a^{7} + \frac{12}{31} a^{6} - \frac{3}{31} a^{5} - \frac{7}{31} a^{4} + \frac{1}{31} a^{3} - \frac{4}{31} a^{2} - \frac{6}{31} a + \frac{3}{31}$
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: | \( 2 a^{6} - 6 a^{5} + 37 a^{4} - 64 a^{3} + 143 a^{2} - 112 a + 53 \), \( \frac{442}{31} a^{7} - \frac{1547}{31} a^{6} + \frac{8811}{31} a^{5} - \frac{18160}{31} a^{4} + \frac{38293}{31} a^{3} - \frac{40053}{31} a^{2} + \frac{23140}{31} a - \frac{5463}{31} \), \( \frac{506}{31} a^{7} - \frac{1771}{31} a^{6} + \frac{10107}{31} a^{5} - \frac{20840}{31} a^{4} + \frac{44185}{31} a^{3} - \frac{46323}{31} a^{2} + \frac{27654}{31} a - \frac{6759}{31} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 93.7356023859 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\times C_4$ (as 8T2):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| An abelian group of order 8 |
| The 8 conjugacy class representatives for $C_4\times C_2$ |
| Character table for $C_4\times C_2$ |
Intermediate fields
| \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{2}, \sqrt{-7})\), 4.0.2048.2, 4.4.100352.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.11.2 | $x^{4} + 8 x + 14$ | $4$ | $1$ | $11$ | $C_4$ | $[3, 4]$ |
| 2.4.11.2 | $x^{4} + 8 x + 14$ | $4$ | $1$ | $11$ | $C_4$ | $[3, 4]$ | |
| $7$ | 7.2.1.2 | $x^{2} + 14$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 7.2.1.2 | $x^{2} + 14$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 14$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.2.1.2 | $x^{2} + 14$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |