Normalized defining polynomial
\( x^{16} - 88 x^{12} - 8384 x^{10} - 2732 x^{8} - 76160 x^{6} + 409360 x^{4} + 106624 x^{2} + 3332 \)
Invariants
| Degree: | $16$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 6]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(890533809374421624858329986629632=2^{64}\cdot 7^{6}\cdot 17^{7}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $114.64$ | 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: | $2, 7, 17$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{16} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} + \frac{1}{8}$, $\frac{1}{16} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} + \frac{1}{8} a$, $\frac{1}{16} a^{10} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} + \frac{1}{8} a^{2}$, $\frac{1}{16} a^{11} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{336} a^{12} - \frac{1}{48} a^{10} + \frac{5}{168} a^{8} - \frac{31}{84} a^{6} - \frac{5}{56} a^{4} - \frac{3}{8} a^{2} + \frac{1}{12}$, $\frac{1}{336} a^{13} - \frac{1}{48} a^{11} + \frac{5}{168} a^{9} - \frac{31}{84} a^{7} - \frac{5}{56} a^{5} - \frac{3}{8} a^{3} + \frac{1}{12} a$, $\frac{1}{1315654785973497648} a^{14} - \frac{112704936493475}{187950683710499664} a^{12} - \frac{19209478538719877}{657827392986748824} a^{10} - \frac{3563184287303009}{1315654785973497648} a^{8} - \frac{87308591218527631}{657827392986748824} a^{6} - \frac{2375369579162143}{31325113951749944} a^{4} - \frac{17931486635815031}{46987670927624916} a^{2} - \frac{18658841875621355}{93975341855249832}$, $\frac{1}{2631309571946995296} a^{15} + \frac{1563352344257209}{1315654785973497648} a^{13} + \frac{2049999042265331}{328913696493374412} a^{11} - \frac{3331086855068513}{187950683710499664} a^{9} - \frac{164845024197092301}{438551595324499216} a^{7} + \frac{64125532486354245}{219275797662249608} a^{5} - \frac{3092284451954401}{46987670927624916} a^{3} - \frac{3762413519931683}{31325113951749944} a$
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
| Rank: | $9$ | 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: | \( 16167613849.8 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A solvable group of order 4096 |
| The 88 conjugacy class representatives for t16n1574 are not computed |
| Character table for t16n1574 is not computed |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.4.243712.1, 8.4.4038896648192.14 |
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 | ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ | $16$ | R | $16$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 | Data not computed | ||||||
| $7$ | 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 7.4.3.1 | $x^{4} + 14$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 7.4.3.1 | $x^{4} + 14$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 7.4.0.1 | $x^{4} + x^{2} - 3 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| $17$ | $\Q_{17}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{17}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 17.8.7.8 | $x^{8} + 4131$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ | |