Normalized defining polynomial
\( x^{16} - 4 x^{15} + 6 x^{14} - x^{13} - 44 x^{12} + 132 x^{11} - 117 x^{10} - 18 x^{9} + 122 x^{8} - 121 x^{7} - 228 x^{6} + 431 x^{5} + 211 x^{4} - 607 x^{3} + 224 x^{2} + 33 x - 19 \)
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: | \(12874930290062500000000=2^{8}\cdot 5^{12}\cdot 11^{8}\cdot 31^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $24.09$ | 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: | $2, 5, 11, 31$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{12} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{45175071510549303} a^{15} - \frac{963892593084134}{15058357170183101} a^{14} - \frac{1815306697524983}{45175071510549303} a^{13} - \frac{2694216609637314}{15058357170183101} a^{12} + \frac{10060556262953762}{45175071510549303} a^{11} + \frac{16995456012232202}{45175071510549303} a^{10} - \frac{14485570831737098}{45175071510549303} a^{9} - \frac{4268380879010804}{15058357170183101} a^{8} - \frac{1029899102548624}{15058357170183101} a^{7} - \frac{4921365271084672}{45175071510549303} a^{6} - \frac{4743286633490146}{45175071510549303} a^{5} - \frac{5137760676228022}{15058357170183101} a^{4} - \frac{6828196715848356}{15058357170183101} a^{3} - \frac{19904616661313410}{45175071510549303} a^{2} + \frac{4435592890974499}{45175071510549303} a - \frac{4923375976397021}{15058357170183101}$
Class group and class number
Trivial group, which has order $1$ (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: | \( 129163.297483 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 1024 |
| The 97 conjugacy class representatives for t16n1086 are not computed |
| Character table for t16n1086 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.15125.1, 4.2.1375.1, 4.2.275.1, 8.4.228765625.1 |
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 | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ | R | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.0.1 | $x^{8} + x^{4} + x^{3} + x + 1$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |
| 2.8.8.5 | $x^{8} + 2 x^{7} + 8 x^{2} + 16$ | $2$ | $4$ | $8$ | $C_8:C_2$ | $[2, 2]^{4}$ | |
| $5$ | 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $11$ | 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.8.4.1 | $x^{8} + 484 x^{4} - 1331 x^{2} + 58564$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $31$ | 31.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 31.2.0.1 | $x^{2} - x + 12$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 31.4.0.1 | $x^{4} - 2 x + 17$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 31.4.2.1 | $x^{4} + 713 x^{2} + 138384$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 31.4.0.1 | $x^{4} - 2 x + 17$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |