Normalized defining polynomial
\( x^{16} - 6 x^{15} + 29 x^{14} - 126 x^{13} + 456 x^{12} - 1068 x^{11} + 1381 x^{10} + 452 x^{9} - 5993 x^{8} + 12728 x^{7} - 11859 x^{6} - 2852 x^{5} + 23536 x^{4} - 32314 x^{3} + 23529 x^{2} - 9214 x + 1601 \)
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: | \(144765030400000000000000=2^{24}\cdot 5^{14}\cdot 29^{2}\cdot 41^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $28.02$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{209} a^{14} + \frac{7}{209} a^{13} + \frac{7}{209} a^{12} + \frac{10}{209} a^{11} + \frac{4}{209} a^{10} - \frac{56}{209} a^{9} - \frac{8}{209} a^{8} - \frac{12}{209} a^{7} - \frac{20}{209} a^{6} + \frac{30}{209} a^{5} - \frac{13}{209} a^{4} + \frac{68}{209} a^{3} - \frac{27}{209} a^{2} - \frac{12}{209} a + \frac{90}{209}$, $\frac{1}{10680568085806740735075649} a^{15} - \frac{14660755753587606027206}{10680568085806740735075649} a^{14} - \frac{692697133213172000502915}{10680568085806740735075649} a^{13} + \frac{3980895766271671316290833}{10680568085806740735075649} a^{12} + \frac{5263791523415064459800547}{10680568085806740735075649} a^{11} - \frac{4930319056513918902399464}{10680568085806740735075649} a^{10} - \frac{977155508225814783422409}{10680568085806740735075649} a^{9} + \frac{109665754788381657440425}{10680568085806740735075649} a^{8} + \frac{3186234440848624978671225}{10680568085806740735075649} a^{7} + \frac{2340162688652860988048576}{10680568085806740735075649} a^{6} - \frac{2768765397988536486959140}{10680568085806740735075649} a^{5} + \frac{3995178736525481351382322}{10680568085806740735075649} a^{4} - \frac{26314944647280471409657}{344534454380862604357279} a^{3} + \frac{3041312790497316071409715}{10680568085806740735075649} a^{2} - \frac{76061049880250288013077}{562135162410881091319771} a + \frac{581928548920270243299318}{10680568085806740735075649}$
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{3605810421858426618496}{970960735073340066825059} a^{15} - \frac{16645484071840816660954}{970960735073340066825059} a^{14} + \frac{3990548966238536238104}{51103196582807371938161} a^{13} - \frac{322766568165687525383234}{970960735073340066825059} a^{12} + \frac{1075864338548349516693524}{970960735073340066825059} a^{11} - \frac{1841532452645842836628197}{970960735073340066825059} a^{10} + \frac{690476668204803401872494}{970960735073340066825059} a^{9} + \frac{5641186510258116410239253}{970960735073340066825059} a^{8} - \frac{15237813717510498710204602}{970960735073340066825059} a^{7} + \frac{16365244174083647328586799}{970960735073340066825059} a^{6} + \frac{4343749398230311482624562}{970960735073340066825059} a^{5} - \frac{32574518841730598208174586}{970960735073340066825059} a^{4} + \frac{64757239547139722280758}{1648490212348624901231} a^{3} - \frac{16103852167836267639732465}{970960735073340066825059} a^{2} - \frac{1078801972069105219981968}{970960735073340066825059} a + \frac{2774253280380229799228422}{970960735073340066825059} \) (order $10$) | 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: | \( 192866.299609 \) (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 t16n797 are not computed |
| Character table for t16n797 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.0.8000.2, \(\Q(\zeta_{5})\), \(\Q(\sqrt{2}, \sqrt{5})\), 8.0.64000000.2 |
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 | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.12.1 | $x^{8} + 6 x^{6} + 8 x^{5} + 16$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ |
| 2.8.12.1 | $x^{8} + 6 x^{6} + 8 x^{5} + 16$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
| 5 | Data not computed | ||||||
| $29$ | 29.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 29.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.2.2 | $x^{4} - 29 x^{2} + 2523$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 41 | Data not computed | ||||||