Normalized defining polynomial
\( x^{16} - 5 x^{15} + 41 x^{14} - 122 x^{13} + 442 x^{12} - 519 x^{11} + 1666 x^{10} - 1132 x^{9} + 7349 x^{8} + 317 x^{7} + 8886 x^{6} + 35394 x^{5} + 21212 x^{4} + 34872 x^{3} + 90206 x^{2} + 80020 x + 23431 \)
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: | \(1855320212000952785484765625=5^{8}\cdot 29^{6}\cdot 41^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $50.61$ | 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: | $5, 29, 41$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{19} a^{14} - \frac{4}{19} a^{13} + \frac{7}{19} a^{11} - \frac{7}{19} a^{10} - \frac{6}{19} a^{9} + \frac{2}{19} a^{7} - \frac{2}{19} a^{6} - \frac{6}{19} a^{5} + \frac{5}{19} a^{4} - \frac{4}{19} a^{3} + \frac{9}{19} a^{2} - \frac{7}{19} a - \frac{4}{19}$, $\frac{1}{75208700055364490206537822746589549} a^{15} - \frac{1402764309914943937474314090330587}{75208700055364490206537822746589549} a^{14} + \frac{34702562039920298900310162155835577}{75208700055364490206537822746589549} a^{13} + \frac{2154256801376359581563559142026435}{75208700055364490206537822746589549} a^{12} - \frac{960063548232157835137695741284357}{3958352634492867905607253828767871} a^{11} + \frac{36847930085093071486691985399325223}{75208700055364490206537822746589549} a^{10} + \frac{34929685531288548384226231418052102}{75208700055364490206537822746589549} a^{9} + \frac{22816948594589852081877488837720447}{75208700055364490206537822746589549} a^{8} + \frac{1045680585316109572151639939610434}{3958352634492867905607253828767871} a^{7} - \frac{1211999228251703530309005035206973}{75208700055364490206537822746589549} a^{6} - \frac{13844437903800014126384821534550152}{75208700055364490206537822746589549} a^{5} + \frac{17556250918577363101995472092980831}{75208700055364490206537822746589549} a^{4} - \frac{17690027470853219793771652404710750}{75208700055364490206537822746589549} a^{3} + \frac{12128261323425170116749121743012192}{75208700055364490206537822746589549} a^{2} - \frac{9949326333338918179227812840453723}{75208700055364490206537822746589549} a + \frac{21659807931319771437252345820517069}{75208700055364490206537822746589549}$
Class group and class number
$C_{4}$, which has order $4$ (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: | \( 2141293.75413 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^2.C_2^2:D_4$ (as 16T225):
| A solvable group of order 128 |
| The 32 conjugacy class representatives for $C_2^2.C_2^2:D_4$ |
| Character table for $C_2^2.C_2^2:D_4$ 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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 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 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $29$ | 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.2.2 | $x^{4} - 29 x^{2} + 2523$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| $41$ | 41.4.2.2 | $x^{4} - 41 x^{2} + 20172$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 41.4.3.3 | $x^{4} + 246$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 41.4.0.1 | $x^{4} - x + 17$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 41.4.3.3 | $x^{4} + 246$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |