Normalized defining polynomial
\( x^{16} + 6 x^{14} - 2 x^{13} + 26 x^{12} - 14 x^{11} + 59 x^{10} - 26 x^{9} + 75 x^{8} - 26 x^{7} + 59 x^{6} - 14 x^{5} + 26 x^{4} - 2 x^{3} + 6 x^{2} + 1 \)
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: | \(2845689369749094400=2^{24}\cdot 5^{2}\cdot 7^{4}\cdot 41^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $14.24$ | 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: | $2, 5, 7, 41$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{35} a^{14} + \frac{12}{35} a^{13} + \frac{9}{35} a^{12} - \frac{11}{35} a^{11} - \frac{2}{7} a^{10} + \frac{17}{35} a^{9} - \frac{1}{5} a^{8} + \frac{13}{35} a^{7} - \frac{1}{5} a^{6} + \frac{17}{35} a^{5} - \frac{2}{7} a^{4} - \frac{11}{35} a^{3} + \frac{9}{35} a^{2} + \frac{12}{35} a + \frac{1}{35}$, $\frac{1}{245} a^{15} - \frac{1}{245} a^{14} - \frac{6}{35} a^{13} - \frac{58}{245} a^{12} - \frac{16}{35} a^{11} - \frac{2}{5} a^{10} - \frac{88}{245} a^{9} - \frac{36}{245} a^{8} - \frac{36}{245} a^{7} + \frac{108}{245} a^{6} + \frac{1}{5} a^{5} + \frac{12}{35} a^{4} - \frac{58}{245} a^{3} + \frac{3}{7} a^{2} - \frac{10}{49} a - \frac{48}{245}$
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: | \( -1 \) (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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 361.678764098 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 16384 |
| The 148 conjugacy class representatives for t16n1772 are not computed |
| Character table for t16n1772 is not computed |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.4.2624.1, 8.2.48197632.1 |
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 | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | R | R | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$ |
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 | ||||||
| $5$ | 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 5.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 5.4.2.2 | $x^{4} - 5 x^{2} + 50$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 5.8.0.1 | $x^{8} + x^{2} - 2 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| $7$ | 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.2.1.2 | $x^{2} + 14$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 7.4.3.2 | $x^{4} - 7$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 7.8.0.1 | $x^{8} - x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| $41$ | 41.4.0.1 | $x^{4} - x + 17$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 41.4.0.1 | $x^{4} - x + 17$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 41.8.4.1 | $x^{8} + 57154 x^{4} - 68921 x^{2} + 816644929$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |