Normalized defining polynomial
\( x^{16} - 26 x^{14} - 234 x^{12} + 5216 x^{10} - 7172 x^{8} - 18512 x^{6} + 5264 x^{4} + 17088 x^{2} + 5696 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[8, 4]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(34608015437945420487065600000000=2^{34}\cdot 5^{8}\cdot 31^{4}\cdot 89^{5}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $93.58$ | 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: | $2, 5, 31, 89$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{4} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{160} a^{12} - \frac{1}{20} a^{10} - \frac{3}{80} a^{8} - \frac{7}{40} a^{6} + \frac{3}{40} a^{4} - \frac{9}{20} a^{2} - \frac{3}{10}$, $\frac{1}{320} a^{13} - \frac{1}{40} a^{11} - \frac{3}{160} a^{9} + \frac{13}{80} a^{7} + \frac{3}{80} a^{5} + \frac{11}{40} a^{3} + \frac{7}{20} a$, $\frac{1}{6126924984640} a^{14} - \frac{641753293}{1531731246160} a^{12} - \frac{100657237967}{3063462492320} a^{10} - \frac{155652152261}{1531731246160} a^{8} - \frac{344903219949}{1531731246160} a^{6} + \frac{32690281725}{153173124616} a^{4} + \frac{7312417}{27758812} a^{2} + \frac{116812002}{1075653965}$, $\frac{1}{12253849969280} a^{15} - \frac{641753293}{3063462492320} a^{13} + \frac{282275573573}{6126924984640} a^{11} + \frac{227280659279}{3063462492320} a^{9} + \frac{38029591591}{3063462492320} a^{7} + \frac{32690281725}{306346249232} a^{5} + \frac{21191823}{55517624} a^{3} - \frac{958841963}{2151307930} a$
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
| Rank: | $11$ | 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: | \( 4878782256.59 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 4096 |
| The 94 conjugacy class representatives for t16n1581 are not computed |
| Character table for t16n1581 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.1103600.2, 4.4.12400.1, 4.4.2225.1, 8.8.1217932960000.1 |
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 | ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.16.34 | $x^{8} + 2 x^{6} + 4 x^{5} + 2 x^{4} + 4$ | $4$ | $2$ | $16$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $[2, 2, 2, 3]^{4}$ |
| 2.8.18.22 | $x^{8} + 2 x^{6} + 2 x^{4} + 20$ | $4$ | $2$ | $18$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $[2, 2, 3, 7/2, 7/2]^{2}$ | |
| $5$ | 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $31$ | 31.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 31.2.1.2 | $x^{2} + 217$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.1.2 | $x^{2} + 217$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.4.2.1 | $x^{4} + 713 x^{2} + 138384$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 31.4.0.1 | $x^{4} - 2 x + 17$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| $89$ | 89.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 89.2.0.1 | $x^{2} - x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 89.2.1.2 | $x^{2} + 267$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 89.2.1.2 | $x^{2} + 267$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 89.4.3.3 | $x^{4} + 267$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 89.4.0.1 | $x^{4} - x + 27$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |