magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![17, 68, 102, -68, -377, -96, 582, 216, -554, -136, 344, 4, -122, 28, 16, -8, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 16*x^14 + 28*x^13 - 122*x^12 + 4*x^11 + 344*x^10 - 136*x^9 - 554*x^8 + 216*x^7 + 582*x^6 - 96*x^5 - 377*x^4 - 68*x^3 + 102*x^2 + 68*x + 17)
gp: K = bnfinit(x^16 - 8*x^15 + 16*x^14 + 28*x^13 - 122*x^12 + 4*x^11 + 344*x^10 - 136*x^9 - 554*x^8 + 216*x^7 + 582*x^6 - 96*x^5 - 377*x^4 - 68*x^3 + 102*x^2 + 68*x + 17, 1)
Normalized defining polynomial
\( x^{16} - 8 x^{15} + 16 x^{14} + 28 x^{13} - 122 x^{12} + 4 x^{11} + 344 x^{10} - 136 x^{9} - 554 x^{8} + 216 x^{7} + 582 x^{6} - 96 x^{5} - 377 x^{4} - 68 x^{3} + 102 x^{2} + 68 x + 17 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(6946827684441348898816=2^{32}\cdot 17^{6}\cdot 113\cdot 593\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $23.18$ | 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: | $2, 17, 113, 593$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} + \frac{3}{7} a^{14} - \frac{3}{7} a^{11} - \frac{1}{7} a^{10} - \frac{3}{7} a^{9} - \frac{1}{7} a^{8} + \frac{2}{7} a^{7} + \frac{1}{7} a^{5} - \frac{1}{7} a^{4} - \frac{3}{7} a^{3} - \frac{3}{7} a^{2} - \frac{1}{7} a + \frac{1}{7}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
Trivial group, which has order $1$ (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: | $11$ | 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: | Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 89836.9337808 \) (assuming GRH) | 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 16384 |
| The 136 conjugacy class representatives for t16n1775 are not computed |
| Character table for t16n1775 is not computed |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.4.4352.1, 8.4.321978368.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
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 | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ | $16$ | ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ | $16$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ | $16$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$ |
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 | Data not computed | ||||||
| $17$ | 17.4.3.4 | $x^{4} + 459$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 17.4.3.4 | $x^{4} + 459$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 17.8.0.1 | $x^{8} + x^{2} - 3 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| $113$ | $\Q_{113}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{113}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 113.2.1.2 | $x^{2} + 339$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 113.2.0.1 | $x^{2} - x + 10$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 113.2.0.1 | $x^{2} - x + 10$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 113.8.0.1 | $x^{8} - 2 x + 5$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 593 | Data not computed | ||||||