Normalized defining polynomial
\( x^{16} - 6 x^{15} + 13 x^{14} - 2 x^{13} - 41 x^{12} + 68 x^{11} - 126 x^{9} + 143 x^{8} + 10 x^{7} - 134 x^{6} + 46 x^{5} + 144 x^{4} - 210 x^{3} + 130 x^{2} - 40 x + 5 \)
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: | \(8861338240000000000=2^{16}\cdot 5^{10}\cdot 61^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $15.28$ | 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: | $2, 5, 61$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{205} a^{14} - \frac{13}{205} a^{13} - \frac{94}{205} a^{12} - \frac{10}{41} a^{11} + \frac{61}{205} a^{10} - \frac{94}{205} a^{9} + \frac{12}{41} a^{8} + \frac{26}{205} a^{7} - \frac{29}{205} a^{6} - \frac{3}{41} a^{5} - \frac{27}{205} a^{4} - \frac{15}{41} a^{3} + \frac{14}{41} a^{2} + \frac{1}{41} a - \frac{6}{41}$, $\frac{1}{5945} a^{15} - \frac{4}{5945} a^{14} - \frac{1851}{5945} a^{13} + \frac{2589}{5945} a^{12} + \frac{1251}{5945} a^{11} + \frac{50}{1189} a^{10} + \frac{239}{5945} a^{9} + \frac{2411}{5945} a^{8} + \frac{3}{29} a^{7} + \frac{544}{5945} a^{6} + \frac{1273}{5945} a^{5} + \frac{707}{5945} a^{4} - \frac{367}{1189} a^{3} - \frac{283}{1189} a^{2} - \frac{366}{1189} a - \frac{218}{1189}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{1500}{41} a^{15} + \frac{8042}{41} a^{14} - \frac{14348}{41} a^{13} - \frac{6239}{41} a^{12} + \frac{57624}{41} a^{11} - \frac{65077}{41} a^{10} - \frac{42089}{41} a^{9} + \frac{162523}{41} a^{8} - \frac{110088}{41} a^{7} - \frac{86665}{41} a^{6} + \frac{146155}{41} a^{5} + \frac{25437}{41} a^{4} - \frac{200745}{41} a^{3} + \frac{186125}{41} a^{2} - \frac{74405}{41} a + \frac{11353}{41} \) (order $4$) | 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: | \( 1649.96109527 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^4.C_2^3$ (as 16T217):
| A solvable group of order 128 |
| The 32 conjugacy class representatives for $C_2^4.C_2^3$ |
| Character table for $C_2^4.C_2^3$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(i, \sqrt{5})\), 8.0.48800000.2, 8.0.48800000.1, 8.0.595360000.5 |
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 | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.8.1 | $x^{8} + 28 x^{4} + 144$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ |
| 2.8.8.1 | $x^{8} + 28 x^{4} + 144$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
| $5$ | 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $61$ | 61.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 61.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 61.4.2.1 | $x^{4} + 183 x^{2} + 14884$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 61.4.2.1 | $x^{4} + 183 x^{2} + 14884$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |