Normalized defining polynomial
\( x^{16} - 7 x^{15} + 36 x^{14} - 143 x^{13} + 326 x^{12} - 401 x^{11} + 44 x^{10} + 2041 x^{9} - 5608 x^{8} + 7716 x^{7} - 2636 x^{6} - 8251 x^{5} + 18471 x^{4} - 24778 x^{3} + 25706 x^{2} - 11857 x + 3191 \)
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: | \(61552138713495855712890625=5^{14}\cdot 11^{8}\cdot 19^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $40.91$ | 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: | $5, 11, 19$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{71} a^{14} + \frac{12}{71} a^{13} - \frac{10}{71} a^{12} + \frac{13}{71} a^{10} - \frac{12}{71} a^{9} + \frac{17}{71} a^{8} - \frac{28}{71} a^{7} - \frac{6}{71} a^{6} + \frac{9}{71} a^{5} + \frac{31}{71} a^{4} + \frac{25}{71} a^{3} + \frac{15}{71} a^{2} - \frac{32}{71} a - \frac{28}{71}$, $\frac{1}{405610506373400375611952108932171} a^{15} + \frac{2316187205347671859792206867770}{405610506373400375611952108932171} a^{14} - \frac{116548982671997337395270780647584}{405610506373400375611952108932171} a^{13} + \frac{60347333800329569235597332381395}{405610506373400375611952108932171} a^{12} - \frac{200003091465842618834124408825380}{405610506373400375611952108932171} a^{11} - \frac{81596339502095429189915810548133}{405610506373400375611952108932171} a^{10} - \frac{147987888337013131303588308594144}{405610506373400375611952108932171} a^{9} + \frac{1505322539136506769407617003085}{13986569185289668124550072721799} a^{8} - \frac{158985372371091865192598796995113}{405610506373400375611952108932171} a^{7} - \frac{118796137181697838075771447263734}{405610506373400375611952108932171} a^{6} + \frac{200887943855697551893360468443857}{405610506373400375611952108932171} a^{5} + \frac{64607392514514658303200456669768}{405610506373400375611952108932171} a^{4} + \frac{29155595584933312641481832949594}{405610506373400375611952108932171} a^{3} - \frac{115630259573193104261640391231900}{405610506373400375611952108932171} a^{2} + \frac{141522755662782951124774065784994}{405610506373400375611952108932171} a - \frac{177839143698340417097038699734914}{405610506373400375611952108932171}$
Class group and class number
$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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: | \( 126840.663141 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^3.C_4$ (as 16T41):
| A solvable group of order 32 |
| The 11 conjugacy class representatives for $C_2^3.C_4$ |
| Character table for $C_2^3.C_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.15125.1, 4.0.5225.1, 4.0.26125.1, 8.4.412921953125.1 x2, 8.0.82584390625.6 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | data not computed |
| Degree 8 siblings: | data not computed |
| Degree 16 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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| 5 | Data not computed | ||||||
| $11$ | 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $19$ | 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 19.2.1.2 | $x^{2} + 76$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.2.0.1 | $x^{2} - x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 19.2.1.2 | $x^{2} + 76$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |