Normalized defining polynomial
\( x^{16} - 5 x^{15} + 48 x^{13} - 96 x^{12} - 91 x^{11} + 569 x^{10} - 402 x^{9} - 1519 x^{8} + 2935 x^{7} + 873 x^{6} - 7511 x^{5} + 8017 x^{4} - 2700 x^{3} - 180 x^{2} - 224 x + 304 \)
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: | \(9756589522702500000000=2^{8}\cdot 3^{8}\cdot 5^{10}\cdot 29^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $23.68$ | 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: | $2, 3, 5, 29$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{14} + \frac{1}{8} a^{13} + \frac{1}{4} a^{12} - \frac{3}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{2} a^{7} - \frac{3}{8} a^{6} - \frac{3}{8} a^{5} + \frac{3}{8} a^{4} - \frac{1}{8} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{3876048420549269584} a^{15} - \frac{222004941446764685}{3876048420549269584} a^{14} - \frac{160558525908369771}{969012105137317396} a^{13} - \frac{106002089994567261}{969012105137317396} a^{12} + \frac{223830366518749813}{484506052568658698} a^{11} + \frac{1721102905311553557}{3876048420549269584} a^{10} + \frac{1502358114922888657}{3876048420549269584} a^{9} + \frac{519509473280725081}{1938024210274634792} a^{8} - \frac{1477146072159281275}{3876048420549269584} a^{7} - \frac{1272178468119642257}{3876048420549269584} a^{6} - \frac{1212589116231042115}{3876048420549269584} a^{5} + \frac{445904591722037645}{3876048420549269584} a^{4} + \frac{714402831689091661}{3876048420549269584} a^{3} - \frac{114634578126076589}{242253026284329349} a^{2} - \frac{81369626400726113}{242253026284329349} a + \frac{75484388800926627}{484506052568658698}$
Class group and class number
$C_{2}$, which has order $2$
Unit group
| Rank: | $7$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{312121137189}{1122413477524} a^{15} + \frac{553924170213}{561206738762} a^{14} + \frac{1601140892135}{1122413477524} a^{13} - \frac{3160556227913}{280603369381} a^{12} + \frac{2915958718902}{280603369381} a^{11} + \frac{45116223298551}{1122413477524} a^{10} - \frac{28012903547222}{280603369381} a^{9} - \frac{36227998049453}{1122413477524} a^{8} + \frac{420028169727149}{1122413477524} a^{7} - \frac{154075894433641}{561206738762} a^{6} - \frac{356509381114261}{561206738762} a^{5} + \frac{326964362637720}{280603369381} a^{4} - \frac{154373657280184}{280603369381} a^{3} - \frac{37191098358659}{1122413477524} a^{2} - \frac{861961551123}{280603369381} a + \frac{16617057276286}{280603369381} \) (order $6$) | 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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 36107.9366643 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^3.C_2^4.C_2$ (as 16T456):
| A solvable group of order 256 |
| The 46 conjugacy class representatives for $C_2^3.C_2^4.C_2$ |
| Character table for $C_2^3.C_2^4.C_2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), 4.0.6525.1, 4.4.725.1, \(\Q(\sqrt{-3}, \sqrt{5})\), 8.0.42575625.1 |
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 | R | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.4.0.1 | $x^{4} - x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 2.4.0.1 | $x^{4} - x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 2.8.8.1 | $x^{8} + 28 x^{4} + 144$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $[2]^{4}$ | |
| $3$ | 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $5$ | 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $29$ | 29.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 29.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.4.2.1 | $x^{4} + 145 x^{2} + 7569$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.1 | $x^{4} + 145 x^{2} + 7569$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.2.1 | $x^{4} + 145 x^{2} + 7569$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |