magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, 1, -8, 27, -32, -8, 78, -107, 54, 35, -84, 77, -46, 20, -6, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 20*x^14 - 46*x^13 + 77*x^12 - 84*x^11 + 35*x^10 + 54*x^9 - 107*x^8 + 78*x^7 - 8*x^6 - 32*x^5 + 27*x^4 - 8*x^3 + x^2 - 2*x + 1)
gp: K = bnfinit(x^16 - 6*x^15 + 20*x^14 - 46*x^13 + 77*x^12 - 84*x^11 + 35*x^10 + 54*x^9 - 107*x^8 + 78*x^7 - 8*x^6 - 32*x^5 + 27*x^4 - 8*x^3 + x^2 - 2*x + 1, 1)
Normalized defining polynomial
\( x^{16} - 6 x^{15} + 20 x^{14} - 46 x^{13} + 77 x^{12} - 84 x^{11} + 35 x^{10} + 54 x^{9} - 107 x^{8} + 78 x^{7} - 8 x^{6} - 32 x^{5} + 27 x^{4} - 8 x^{3} + x^{2} - 2 x + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 8]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(34870903681646592=2^{16}\cdot 3^{13}\cdot 7^{4}\cdot 139\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $10.81$ | 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: | $2, 3, 7, 139$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{28487} a^{15} - \frac{8298}{28487} a^{14} + \frac{10931}{28487} a^{13} + \frac{5736}{28487} a^{12} + \frac{10455}{28487} a^{11} - \frac{7003}{28487} a^{10} + \frac{12405}{28487} a^{9} + \frac{4351}{28487} a^{8} - \frac{14057}{28487} a^{7} - \frac{8082}{28487} a^{6} - \frac{13975}{28487} a^{5} - \frac{4448}{28487} a^{4} - \frac{7822}{28487} a^{3} - \frac{4883}{28487} a^{2} + \frac{9810}{28487} a - \frac{14137}{28487}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
Trivial group, which has order $1$
magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp
Unit group
magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{7372}{467} a^{15} - \frac{39287}{467} a^{14} + \frac{121100}{467} a^{13} - \frac{257908}{467} a^{12} + \frac{394728}{467} a^{11} - \frac{354653}{467} a^{10} + \frac{20400}{467} a^{9} + \frac{411571}{467} a^{8} - \frac{512736}{467} a^{7} + \frac{231922}{467} a^{6} + \frac{94570}{467} a^{5} - \frac{170239}{467} a^{4} + \frac{84502}{467} a^{3} - \frac{3451}{467} a^{2} + \frac{6238}{467} a - \frac{10650}{467} \) (order $6$) | 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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 120.015313284 \) | 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 8192 |
| The 152 conjugacy class representatives for t16n1719 are not computed |
| Character table for t16n1719 is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 4.0.189.1, 4.0.1008.1, 4.0.432.1, 8.0.9144576.2 |
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 | R | ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| $3$ | 3.8.7.1 | $x^{8} + 3$ | $8$ | $1$ | $7$ | $QD_{16}$ | $[\ ]_{8}^{2}$ |
| 3.8.6.2 | $x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ | |
| $7$ | 7.4.0.1 | $x^{4} + x^{2} - 3 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 7.4.0.1 | $x^{4} + x^{2} - 3 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $139$ | $\Q_{139}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{139}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{139}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{139}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 139.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 139.2.1.2 | $x^{2} + 556$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 139.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 139.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 139.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |