Normalized defining polynomial
\( x^{16} - 8 x^{15} + 22 x^{14} - 14 x^{13} - 33 x^{12} + 16 x^{11} + 149 x^{10} - 272 x^{9} + 148 x^{8} + 72 x^{7} - 4 x^{6} - 339 x^{5} + 613 x^{4} - 563 x^{3} + 283 x^{2} - 71 x + 9 \)
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: | \(1166177856221725962879121=11^{4}\cdot 13^{12}\cdot 43^{4}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $31.93$ | 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: | $11, 13, 43$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{273781} a^{14} - \frac{7}{273781} a^{13} - \frac{51983}{273781} a^{12} + \frac{38208}{273781} a^{11} + \frac{10396}{273781} a^{10} + \frac{2315}{6367} a^{9} - \frac{27820}{273781} a^{8} - \frac{65416}{273781} a^{7} + \frac{134069}{273781} a^{6} - \frac{93616}{273781} a^{5} - \frac{127879}{273781} a^{4} - \frac{109630}{273781} a^{3} + \frac{24078}{273781} a^{2} - \frac{103727}{273781} a - \frac{110775}{273781}$, $\frac{1}{105953247} a^{15} + \frac{62}{35317749} a^{14} - \frac{15385070}{105953247} a^{13} - \frac{5339231}{35317749} a^{12} - \frac{4229234}{11772583} a^{11} + \frac{26198701}{105953247} a^{10} - \frac{14490698}{105953247} a^{9} - \frac{86711}{11772583} a^{8} - \frac{33024794}{105953247} a^{7} - \frac{38830615}{105953247} a^{6} - \frac{15465070}{35317749} a^{5} - \frac{14925430}{35317749} a^{4} + \frac{44299147}{105953247} a^{3} + \frac{15112232}{35317749} a^{2} - \frac{49698434}{105953247} a + \frac{1822467}{11772583}$
Class group and class number
$C_{2}$, which has order $2$ (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: | \( 621355.668551 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^4.(C_4\times S_3)$ (as 16T725):
| A solvable group of order 384 |
| The 28 conjugacy class representatives for $C_2^4.(C_4\times S_3)$ |
| Character table for $C_2^4.(C_4\times S_3)$ is not computed |
Intermediate fields
| \(\Q(\sqrt{13}) \), 4.0.2197.1, 8.0.1079897150761.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 12 siblings: | data not computed |
| Degree 16 sibling: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 32 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 | ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }$ | ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }$ | R | R | ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $11$ | 11.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
| 11.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 11.8.4.1 | $x^{8} + 484 x^{4} - 1331 x^{2} + 58564$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $13$ | 13.4.3.2 | $x^{4} - 52$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 13.4.3.2 | $x^{4} - 52$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 13.8.6.1 | $x^{8} - 13 x^{4} + 2704$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $43$ | 43.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 43.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 43.4.2.1 | $x^{4} + 215 x^{2} + 16641$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 43.4.2.1 | $x^{4} + 215 x^{2} + 16641$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |