magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12994449551, 0, 198979236, 0, 973314, 0, 1804, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^8 + 1804*x^6 + 973314*x^4 + 198979236*x^2 + 12994449551)
gp: K = bnfinit(x^8 + 1804*x^6 + 973314*x^4 + 198979236*x^2 + 12994449551, 1)
Normalized defining polynomial
\( x^{8} + 1804 x^{6} + 973314 x^{4} + 198979236 x^{2} + 12994449551 \)
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: | \(13952683962766721024=2^{30}\cdot 2351^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $247.22$ | 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: | $2, 2351$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2351} a^{4} - \frac{547}{2351} a^{2}$, $\frac{1}{2351} a^{5} - \frac{547}{2351} a^{3}$, $\frac{1}{15514853207} a^{6} - \frac{374356}{15514853207} a^{4} + \frac{504061}{6599257} a^{2} + \frac{1000}{2807}$, $\frac{1}{15514853207} a^{7} - \frac{374356}{15514853207} a^{5} + \frac{504061}{6599257} a^{3} + \frac{1000}{2807} a$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{2}\times C_{2}\times C_{26016}$, which has order $104064$ (assuming GRH)
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: | \( \frac{188}{2216407601} a^{6} + \frac{327397}{2216407601} a^{4} + \frac{64110}{942751} a^{2} + \frac{2738}{401} \), \( \frac{808}{15514853207} a^{6} + \frac{1086174}{15514853207} a^{4} + \frac{89447}{6599257} a^{2} - \frac{6030}{2807} \), \( \frac{4723}{15514853207} a^{6} + \frac{7116745}{15514853207} a^{4} + \frac{1071116}{6599257} a^{2} + \frac{43731}{2807} \) (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 19.534360053 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2\wr C_4$ (as 8T27):
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A solvable group of order 64 |
| The 13 conjugacy class representatives for $((C_8 : C_2):C_2):C_2$ |
| Character table for $((C_8 : C_2):C_2):C_2$ |
Intermediate fields
| \(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 8 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 32 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 | ${\href{/LocalNumberField/3.8.0.1}{8} }$ | ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }$ | ${\href{/LocalNumberField/13.8.0.1}{8} }$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }$ | ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }$ | ${\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }$ | ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }$ | ${\href{/LocalNumberField/59.8.0.1}{8} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.30.34 | $x^{8} + 8 x^{7} + 8 x^{2} + 30$ | $8$ | $1$ | $30$ | $((C_8 : C_2):C_2):C_2$ | $[2, 3, 7/2, 4, 17/4, 19/4]$ |
| 2351 | Data not computed | ||||||