Normalized defining polynomial
\( x^{16} - 6 x^{15} + 5 x^{14} + 38 x^{13} - 108 x^{12} + 132 x^{11} - 169 x^{10} - 114 x^{9} + 1662 x^{8} - 2822 x^{7} - 44 x^{6} + 3970 x^{5} - 3084 x^{4} - 132 x^{3} + 721 x^{2} + 198 x + 41 \)
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: | \(131954243069542400000000=2^{24}\cdot 5^{8}\cdot 13^{4}\cdot 89^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $27.86$ | 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: | $2, 5, 13, 89$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{17} a^{14} - \frac{2}{17} a^{13} - \frac{6}{17} a^{12} + \frac{3}{17} a^{11} + \frac{7}{17} a^{10} - \frac{2}{17} a^{9} + \frac{6}{17} a^{8} + \frac{1}{17} a^{7} - \frac{1}{17} a^{6} - \frac{7}{17} a^{5} - \frac{1}{17} a^{4} - \frac{8}{17} a^{3} - \frac{2}{17} a^{2} + \frac{3}{17} a + \frac{8}{17}$, $\frac{1}{572700407553777700313} a^{15} - \frac{16796727790461090773}{572700407553777700313} a^{14} + \frac{155955678163406648801}{572700407553777700313} a^{13} - \frac{13876095704268306051}{572700407553777700313} a^{12} + \frac{241302235439530952684}{572700407553777700313} a^{11} + \frac{269468285001961713400}{572700407553777700313} a^{10} - \frac{223905970647460653460}{572700407553777700313} a^{9} - \frac{84183210602893271558}{572700407553777700313} a^{8} + \frac{71823127618793298285}{572700407553777700313} a^{7} + \frac{5180797400912702430}{18474206695283151623} a^{6} - \frac{96311742800449280023}{572700407553777700313} a^{5} - \frac{236874689966128913888}{572700407553777700313} a^{4} - \frac{9987943952603420704}{33688259267869276489} a^{3} + \frac{168416041132468722608}{572700407553777700313} a^{2} + \frac{155325902338926124420}{572700407553777700313} a + \frac{83280865706991805497}{572700407553777700313}$
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: | \( 312858.932494 \) (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 t16n1577 are not computed |
| Character table for t16n1577 is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.8.432640000.1 |
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 | ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{2}$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{6}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$ |
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 | ||||||
| 5 | Data not computed | ||||||
| $13$ | 13.4.2.2 | $x^{4} - 13 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 13.4.2.2 | $x^{4} - 13 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 13.8.0.1 | $x^{8} + 4 x^{2} - x + 6$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
| 89 | Data not computed | ||||||