Normalized defining polynomial
\( x^{16} - 8 x^{15} + 42 x^{14} - 154 x^{13} + 382 x^{12} - 554 x^{11} - 137 x^{10} + 2288 x^{9} - 4195 x^{8} + 5754 x^{7} - 11486 x^{6} + 18476 x^{5} - 17552 x^{4} + 14776 x^{3} - 18480 x^{2} + 14992 x - 3184 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(14754410872643200000000000=2^{16}\cdot 5^{11}\cdot 11^{5}\cdot 31^{5}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $37.42$ | 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: | $2, 5, 11, 31$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{20} a^{12} + \frac{1}{5} a^{11} - \frac{1}{5} a^{10} - \frac{1}{2} a^{9} - \frac{3}{10} a^{8} - \frac{1}{10} a^{7} + \frac{7}{20} a^{6} - \frac{2}{5} a^{5} - \frac{9}{20} a^{4} + \frac{3}{10} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5}$, $\frac{1}{20} a^{13} - \frac{1}{5} a^{10} - \frac{3}{10} a^{9} + \frac{1}{10} a^{8} - \frac{1}{4} a^{7} + \frac{1}{5} a^{6} + \frac{3}{20} a^{5} - \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{3}{10} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{2200} a^{14} - \frac{4}{275} a^{13} - \frac{1}{50} a^{12} - \frac{37}{220} a^{11} - \frac{21}{100} a^{10} + \frac{457}{1100} a^{9} + \frac{27}{88} a^{8} - \frac{67}{550} a^{7} - \frac{13}{2200} a^{6} - \frac{21}{100} a^{5} + \frac{201}{550} a^{4} - \frac{9}{275} a^{3} + \frac{23}{110} a^{2} - \frac{97}{275} a - \frac{137}{275}$, $\frac{1}{11565909966176663840600} a^{15} + \frac{63774485201226797}{889685382013589526200} a^{14} + \frac{19560478396671808361}{1156590996617666384060} a^{13} - \frac{4211897015486873421}{222421345503397381550} a^{12} - \frac{34986575172613149424}{1445738745772082980075} a^{11} - \frac{645467918708831192213}{2891477491544165960150} a^{10} - \frac{1809593478615387308803}{11565909966176663840600} a^{9} + \frac{285838485504477225613}{608732103482982307400} a^{8} + \frac{1588590448368536570973}{11565909966176663840600} a^{7} - \frac{3673380754651453894661}{11565909966176663840600} a^{6} + \frac{667147106091974769796}{1445738745772082980075} a^{5} - \frac{474162839575153420987}{1156590996617666384060} a^{4} - \frac{179155194065960740832}{1445738745772082980075} a^{3} - \frac{655208630955567318662}{1445738745772082980075} a^{2} + \frac{39711004959916252152}{1445738745772082980075} a - \frac{624819327585150335416}{1445738745772082980075}$
Class group and class number
$C_{2}\times C_{2}$, which has order $4$ (assuming GRH)
Unit group
| Rank: | $9$ | 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: | \( 1528295.73077 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 16384 |
| The 130 conjugacy class representatives for t16n1782 are not computed |
| Character table for t16n1782 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.8525.1, 8.6.18604960000.2 |
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 | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/13.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.8.7 | $x^{8} + 2 x^{6} + 4 x^{5} + 16$ | $2$ | $4$ | $8$ | $((C_8 : C_2):C_2):C_2$ | $[2, 2, 2, 2]^{4}$ |
| 2.8.8.7 | $x^{8} + 2 x^{6} + 4 x^{5} + 16$ | $2$ | $4$ | $8$ | $((C_8 : C_2):C_2):C_2$ | $[2, 2, 2, 2]^{4}$ | |
| $5$ | 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.8.6.2 | $x^{8} + 15 x^{4} + 100$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $11$ | $\Q_{11}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{11}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{11}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| $\Q_{11}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.4.3.2 | $x^{4} - 11$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 11.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 31 | Data not computed | ||||||