Normalized defining polynomial
\( x^{16} + 8 x^{14} - 24 x^{13} - 4 x^{12} - 48 x^{11} + 88 x^{10} + 104 x^{9} + 110 x^{8} - 80 x^{7} - 168 x^{6} - 104 x^{5} + 132 x^{4} + 64 x^{3} - 24 x^{2} - 40 x + 25 \)
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: | \(27487790694400000000=2^{46}\cdot 5^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $16.40$ | 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: | $2, 5$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{8} - \frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{9} - \frac{1}{4} a^{5} + \frac{1}{4} a$, $\frac{1}{12} a^{14} + \frac{1}{12} a^{12} + \frac{1}{12} a^{10} - \frac{1}{4} a^{8} + \frac{1}{6} a^{7} - \frac{1}{12} a^{6} + \frac{1}{6} a^{5} - \frac{5}{12} a^{4} + \frac{1}{6} a^{3} - \frac{1}{12} a^{2} + \frac{1}{6} a - \frac{5}{12}$, $\frac{1}{6614056414740} a^{15} + \frac{231244981}{220468547158} a^{14} + \frac{243755630699}{3307028207370} a^{13} + \frac{213708359467}{2204685471580} a^{12} - \frac{19084645717}{3307028207370} a^{11} - \frac{546624089141}{2204685471580} a^{10} - \frac{261106588149}{2204685471580} a^{9} + \frac{11171929136}{53339164635} a^{8} - \frac{277736353643}{1322811282948} a^{7} - \frac{71036461466}{330702820737} a^{6} + \frac{452939450188}{1653514103685} a^{5} - \frac{1839548074399}{6614056414740} a^{4} + \frac{25090413383}{53339164635} a^{3} + \frac{2312673461849}{6614056414740} a^{2} - \frac{2344406818859}{6614056414740} a + \frac{36394799533}{220468547158}$
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{270159}{44076079} a^{15} - \frac{429415}{44076079} a^{14} - \frac{1114447}{44076079} a^{13} + \frac{10334055}{88152158} a^{12} + \frac{17663345}{44076079} a^{11} + \frac{5969315}{44076079} a^{10} - \frac{68446287}{44076079} a^{9} - \frac{2633915}{1421809} a^{8} - \frac{20301767}{44076079} a^{7} + \frac{207652163}{44076079} a^{6} + \frac{232414643}{44076079} a^{5} + \frac{28293737}{88152158} a^{4} - \frac{5465965}{1421809} a^{3} - \frac{39499227}{44076079} a^{2} - \frac{1120645}{44076079} a + \frac{9335578}{44076079} \) (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 (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 3495.44381092 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^2\wr C_2$ (as 16T39):
| A solvable group of order 32 |
| The 14 conjugacy class representatives for $C_2^2\wr C_2$ |
| Character table for $C_2^2\wr C_2$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 8 siblings: | data not computed |
| Degree 16 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 | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 | Data not computed | ||||||
| $5$ | 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |