Normalized defining polynomial
\( x^{16} - 4 x^{15} + 13 x^{14} - 42 x^{13} - 113 x^{12} + 726 x^{11} - 756 x^{10} - 728 x^{9} + 1108 x^{8} + 2649 x^{7} - 7901 x^{6} + 3896 x^{5} + 7426 x^{4} - 4265 x^{3} - 3505 x^{2} + 295 x + 295 \)
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: | \(3032754840090903896728515625=5^{12}\cdot 59^{6}\cdot 131^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $52.19$ | 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: | $5, 59, 131$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{6712983845794471632707899679} a^{15} + \frac{1852241482428848792662585616}{6712983845794471632707899679} a^{14} - \frac{1899327884142142999791462574}{6712983845794471632707899679} a^{13} - \frac{3017212623740505357308677813}{6712983845794471632707899679} a^{12} - \frac{2192670335824141844504441333}{6712983845794471632707899679} a^{11} + \frac{935981184294023022520560870}{6712983845794471632707899679} a^{10} + \frac{2197775990171900867584739342}{6712983845794471632707899679} a^{9} + \frac{579981499737990451731192933}{6712983845794471632707899679} a^{8} - \frac{3154881463984624337425902420}{6712983845794471632707899679} a^{7} + \frac{1539860478348099319282484891}{6712983845794471632707899679} a^{6} + \frac{2171930271199873312564185292}{6712983845794471632707899679} a^{5} - \frac{1678399658662443475380465395}{6712983845794471632707899679} a^{4} + \frac{527703681778705790574409332}{6712983845794471632707899679} a^{3} + \frac{2987864134746317660756710997}{6712983845794471632707899679} a^{2} - \frac{2632207419227872010728819355}{6712983845794471632707899679} a + \frac{1616402773765678606982368048}{6712983845794471632707899679}$
Class group and class number
$C_{2}$, which has order $2$ (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: | \( 46212578.8259 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 2304 |
| The 40 conjugacy class representatives for t16n1496 |
| Character table for t16n1496 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.966125.1, 8.4.16815405625.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 12 siblings: | data not computed |
| Degree 16 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 siblings: | data not computed |
| Degree 36 siblings: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }$ | ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ | R | ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ | ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ | ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ | ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ | ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ | R |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.4.3.1 | $x^{4} - 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 59 | Data not computed | ||||||
| $131$ | 131.2.1.2 | $x^{2} + 393$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 131.2.1.2 | $x^{2} + 393$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 131.2.0.1 | $x^{2} - x + 14$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 131.4.2.1 | $x^{4} + 3537 x^{2} + 3363556$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 131.6.0.1 | $x^{6} - 3 x + 54$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |