Normalized defining polynomial
\( x^{16} + 72 x^{14} + 2226 x^{12} + 32256 x^{10} + 211387 x^{8} + 464304 x^{6} + 1292884 x^{4} - 194208 x^{2} + 56644 \)
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: | \(204626334874637321888403030016=2^{56}\cdot 7^{6}\cdot 17^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $67.91$ | 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: | $2, 7, 17$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{34} a^{12} + \frac{2}{17} a^{10} + \frac{8}{17} a^{8} - \frac{5}{17} a^{6} + \frac{9}{34} a^{4}$, $\frac{1}{68} a^{13} - \frac{15}{34} a^{11} + \frac{4}{17} a^{9} + \frac{6}{17} a^{7} - \frac{25}{68} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{762845811646199302756588} a^{14} + \frac{1582887190347987407297}{381422905823099651378294} a^{12} - \frac{7059092688698732420256}{27244493273078546527021} a^{10} + \frac{4093158100689476105485}{27244493273078546527021} a^{8} - \frac{272845710304137161532921}{762845811646199302756588} a^{6} - \frac{167561306251786430202397}{381422905823099651378294} a^{4} + \frac{4013207791025993622911}{22436641519005861845782} a^{2} - \frac{300010497246956778473}{1602617251357561560413}$, $\frac{1}{762845811646199302756588} a^{15} + \frac{1582887190347987407297}{381422905823099651378294} a^{13} - \frac{7059092688698732420256}{27244493273078546527021} a^{11} + \frac{4093158100689476105485}{27244493273078546527021} a^{9} - \frac{272845710304137161532921}{762845811646199302756588} a^{7} - \frac{167561306251786430202397}{381422905823099651378294} a^{5} + \frac{4013207791025993622911}{22436641519005861845782} a^{3} - \frac{300010497246956778473}{1602617251357561560413} a$
Class group and class number
$C_{2}\times C_{2}$, which has order $4$ (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: | \( 90544245.3735 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^2.C_2^2:D_4$ (as 16T225):
| A solvable group of order 128 |
| The 32 conjugacy class representatives for $C_2^2.C_2^2:D_4$ |
| Character table for $C_2^2.C_2^2:D_4$ is not computed |
Intermediate fields
| \(\Q(\sqrt{2}) \), 4.0.7168.1, 4.4.121856.2, 4.0.1088.2, 8.0.14848884736.10 |
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 | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.28.78 | $x^{8} + 12 x^{4} + 92$ | $8$ | $1$ | $28$ | $Z_8 : Z_8^\times$ | $[2, 3, 7/2, 9/2]^{2}$ |
| 2.8.28.78 | $x^{8} + 12 x^{4} + 92$ | $8$ | $1$ | $28$ | $Z_8 : Z_8^\times$ | $[2, 3, 7/2, 9/2]^{2}$ | |
| $7$ | 7.4.2.2 | $x^{4} - 7 x^{2} + 147$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 7.4.0.1 | $x^{4} + x^{2} - 3 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| $17$ | 17.4.3.4 | $x^{4} + 459$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 17.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 17.4.3.3 | $x^{4} + 51$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 17.4.0.1 | $x^{4} - x + 11$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |