Normalized defining polynomial
\( x^{16} - 111 x^{14} + 5910 x^{12} - 197473 x^{10} + 4571039 x^{8} - 76365265 x^{6} + 911551490 x^{4} - 7174634075 x^{2} + 29723484025 \)
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: | \(1808302579025794791040000000000=2^{16}\cdot 5^{10}\cdot 29^{6}\cdot 41^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $77.82$ | 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: | $2, 5, 29, 41$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{1189} a^{10} + \frac{498}{1189} a^{8} + \frac{371}{1189} a^{6} - \frac{534}{1189} a^{4} + \frac{552}{1189} a^{2}$, $\frac{1}{1189} a^{11} + \frac{498}{1189} a^{9} + \frac{371}{1189} a^{7} - \frac{534}{1189} a^{5} + \frac{552}{1189} a^{3}$, $\frac{1}{172405} a^{12} + \frac{34}{172405} a^{10} - \frac{1196}{34481} a^{8} + \frac{41342}{172405} a^{6} + \frac{15284}{172405} a^{4} + \frac{13}{1189} a^{2}$, $\frac{1}{172405} a^{13} + \frac{34}{172405} a^{11} - \frac{1196}{34481} a^{9} + \frac{41342}{172405} a^{7} + \frac{15284}{172405} a^{5} + \frac{13}{1189} a^{3}$, $\frac{1}{149774141440137187414705} a^{14} + \frac{364487675379457108}{149774141440137187414705} a^{12} - \frac{28381080606945250634}{149774141440137187414705} a^{10} + \frac{19108413887482819924877}{149774141440137187414705} a^{8} - \frac{66553218704298770825713}{149774141440137187414705} a^{6} + \frac{1590731256327771809089}{5164625566901282324645} a^{4} - \frac{3178842189876889552}{35618107357939878101} a^{2} + \frac{14176472975474176}{29956356062186609}$, $\frac{1}{4343450101763978435026445} a^{15} + \frac{2970690652789692091}{4343450101763978435026445} a^{13} - \frac{191703133857986642902}{4343450101763978435026445} a^{11} + \frac{1525576264591549964206672}{4343450101763978435026445} a^{9} - \frac{501597125645805459009632}{4343450101763978435026445} a^{7} + \frac{7603331394281370474697}{149774141440137187414705} a^{5} - \frac{426881542333444287248}{1032925113380256464929} a^{3} + \frac{223870965410780439}{868734325803411661} a$
Class group and class number
$C_{2}\times C_{4}$, which has order $8$ (assuming GRH)
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 (assuming GRH) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 46792538.9982 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 512 |
| The 62 conjugacy class representatives for t16n790 are not computed |
| Character table for t16n790 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.725.1, 4.0.29725.2, 4.0.1025.1, 8.0.883575625.2 |
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 | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.8.6 | $x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$ | $2$ | $4$ | $8$ | $(C_8:C_2):C_2$ | $[2, 2, 2]^{4}$ |
| 2.8.8.6 | $x^{8} + 2 x^{7} + 2 x^{6} + 16 x^{2} + 16$ | $2$ | $4$ | $8$ | $(C_8:C_2):C_2$ | $[2, 2, 2]^{4}$ | |
| $5$ | 5.8.6.2 | $x^{8} + 15 x^{4} + 100$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $29$ | 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.4.2.1 | $x^{4} + 145 x^{2} + 7569$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.1 | $x^{4} + 145 x^{2} + 7569$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $41$ | 41.4.2.2 | $x^{4} - 41 x^{2} + 20172$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 41.4.0.1 | $x^{4} - x + 17$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 41.4.2.1 | $x^{4} + 943 x^{2} + 242064$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 41.4.2.1 | $x^{4} + 943 x^{2} + 242064$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |