Normalized defining polynomial
\( x^{16} - 400 x^{14} + 64435 x^{12} - 5451425 x^{10} + 263794450 x^{8} - 7417493375 x^{6} + 116846137750 x^{4} - 922958901250 x^{2} + 2662128788125 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[16, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(126656373383578014838096000000000000=2^{16}\cdot 5^{12}\cdot 29^{4}\cdot 1621^{5}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $156.28$ | 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: | $2, 5, 29, 1621$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{25} a^{8}$, $\frac{1}{25} a^{9}$, $\frac{1}{125} a^{10} + \frac{1}{25} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{125} a^{11} + \frac{1}{25} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{202625} a^{12} - \frac{16}{8105} a^{10} - \frac{81}{40525} a^{8} + \frac{778}{8105} a^{6} + \frac{689}{8105} a^{4}$, $\frac{1}{202625} a^{13} - \frac{16}{8105} a^{11} - \frac{81}{40525} a^{9} + \frac{778}{8105} a^{7} + \frac{689}{8105} a^{5}$, $\frac{1}{16749066642094455627377786875} a^{14} + \frac{7716078864550529986156}{3349813328418891125475557375} a^{12} - \frac{11560535703897737794728958}{3349813328418891125475557375} a^{10} + \frac{536319348161160053196416}{133992533136755645019022295} a^{8} - \frac{65104708101580869600210508}{669962665683778225095111475} a^{6} - \frac{803864640511560681099}{16532083052036476868479} a^{4} - \frac{6541333024792990680984}{82660415260182384342395} a^{2} + \frac{1148527972639579677}{10198694048140948099}$, $\frac{1}{16749066642094455627377786875} a^{15} + \frac{7716078864550529986156}{3349813328418891125475557375} a^{13} - \frac{11560535703897737794728958}{3349813328418891125475557375} a^{11} + \frac{536319348161160053196416}{133992533136755645019022295} a^{9} - \frac{65104708101580869600210508}{669962665683778225095111475} a^{7} - \frac{803864640511560681099}{16532083052036476868479} a^{5} - \frac{6541333024792990680984}{82660415260182384342395} a^{3} + \frac{1148527972639579677}{10198694048140948099} a$
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
| Rank: | $15$ | 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: | \( 2259683092720 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 2048 |
| The 71 conjugacy class representatives for t16n1430 are not computed |
| Character table for t16n1430 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.725.1, 8.8.852038125.1 |
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 | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | $16$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | R | ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | $16$ | $16$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$ |
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 | ||||||
| $5$ | 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $29$ | 29.4.2.1 | $x^{4} + 145 x^{2} + 7569$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 29.4.2.1 | $x^{4} + 145 x^{2} + 7569$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.8.0.1 | $x^{8} + x^{2} - 3 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 1621 | Data not computed | ||||||