Normalized defining polynomial
\( x^{16} - x^{15} + 6 x^{14} - 18 x^{13} + 22 x^{12} - 100 x^{11} + 118 x^{10} - 258 x^{9} + 472 x^{8} - 348 x^{7} + 1000 x^{6} - 260 x^{5} + 1090 x^{4} - 126 x^{3} + 576 x^{2} - 44 x + 121 \)
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: | \(5012631230712890625=3^{8}\cdot 5^{12}\cdot 29^{2}\cdot 61^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $14.75$ | 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: | $3, 5, 29, 61$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{397260331919} a^{15} - \frac{98485205655}{397260331919} a^{14} + \frac{28788190833}{397260331919} a^{13} + \frac{32062933698}{397260331919} a^{12} - \frac{9589607606}{36114575629} a^{11} - \frac{166461471903}{397260331919} a^{10} - \frac{147202910165}{397260331919} a^{9} + \frac{186523903451}{397260331919} a^{8} - \frac{103273410439}{397260331919} a^{7} + \frac{136700618245}{397260331919} a^{6} - \frac{122008487030}{397260331919} a^{5} - \frac{49682580603}{397260331919} a^{4} + \frac{116356746870}{397260331919} a^{3} - \frac{29413078755}{397260331919} a^{2} - \frac{89091526443}{397260331919} a - \frac{14842349260}{36114575629}$
Class group and class number
Trivial group, which has order $1$
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{121530015249}{397260331919} a^{15} - \frac{29553137355}{397260331919} a^{14} + \frac{406105800383}{397260331919} a^{13} - \frac{1558073336701}{397260331919} a^{12} + \frac{18392320362}{36114575629} a^{11} - \frac{7047531756996}{397260331919} a^{10} + \frac{4331442559385}{397260331919} a^{9} - \frac{6419185742786}{397260331919} a^{8} + \frac{23504506175101}{397260331919} a^{7} + \frac{16669143903542}{397260331919} a^{6} + \frac{39702163496727}{397260331919} a^{5} + \frac{37703902214878}{397260331919} a^{4} + \frac{24507268327008}{397260331919} a^{3} + \frac{25940602843768}{397260331919} a^{2} + \frac{4395937637067}{397260331919} a + \frac{650868025820}{36114575629} \) (order $30$) | 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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 4651.29946037 \) | 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
| \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\zeta_{15})\) |
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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ | R | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||
| 5 | Data not computed | ||||||
| $29$ | 29.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 29.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.4.2.1 | $x^{4} + 145 x^{2} + 7569$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 61 | Data not computed | ||||||