Normalized defining polynomial
\( x^{16} - 8 x^{15} + 27 x^{14} - 49 x^{13} + 53 x^{12} - 45 x^{11} - 33 x^{10} + 363 x^{9} - 484 x^{8} - 451 x^{7} + 1815 x^{6} - 2129 x^{5} + 5031 x^{4} - 7873 x^{3} - 2049 x^{2} + 5831 x + 2107 \)
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: | \(240414920542051180813313277797=11^{10}\cdot 53^{11}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $68.60$ | 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: | $11, 53$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{77} a^{12} - \frac{6}{77} a^{11} + \frac{30}{77} a^{10} - \frac{18}{77} a^{9} - \frac{18}{77} a^{8} + \frac{37}{77} a^{7} + \frac{31}{77} a^{6} + \frac{2}{7} a^{5} + \frac{5}{11} a^{4} - \frac{2}{7} a^{3} + \frac{26}{77} a^{2} + \frac{36}{77} a - \frac{2}{11}$, $\frac{1}{77} a^{13} - \frac{6}{77} a^{11} + \frac{8}{77} a^{10} + \frac{4}{11} a^{9} + \frac{6}{77} a^{8} + \frac{2}{7} a^{7} - \frac{23}{77} a^{6} + \frac{13}{77} a^{5} + \frac{34}{77} a^{4} - \frac{29}{77} a^{3} + \frac{38}{77} a^{2} - \frac{29}{77} a - \frac{1}{11}$, $\frac{1}{477095465} a^{14} - \frac{1}{68156495} a^{13} - \frac{264021}{477095465} a^{12} + \frac{1584217}{477095465} a^{11} - \frac{103081929}{477095465} a^{10} + \frac{23792024}{477095465} a^{9} - \frac{347231}{95419093} a^{8} - \frac{118380851}{477095465} a^{7} - \frac{30768915}{95419093} a^{6} + \frac{19918936}{95419093} a^{5} - \frac{3403819}{8674463} a^{4} + \frac{31447261}{477095465} a^{3} - \frac{138344418}{477095465} a^{2} + \frac{69348353}{477095465} a - \frac{5875074}{68156495}$, $\frac{1}{93987806605} a^{15} + \frac{13}{13426829515} a^{14} - \frac{539320622}{93987806605} a^{13} + \frac{428021444}{93987806605} a^{12} + \frac{2481020977}{93987806605} a^{11} + \frac{41640150597}{93987806605} a^{10} - \frac{10409186323}{93987806605} a^{9} - \frac{46969527071}{93987806605} a^{8} + \frac{4015484927}{8544346055} a^{7} - \frac{789513023}{1708869211} a^{6} + \frac{5842906249}{18797561321} a^{5} + \frac{14015825661}{93987806605} a^{4} + \frac{4110529410}{18797561321} a^{3} - \frac{3580715701}{8544346055} a^{2} - \frac{13332564814}{93987806605} a - \frac{1077636897}{13426829515}$
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: | \( 143505705.722 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_4:D_4.D_4$ (as 16T681):
| A solvable group of order 256 |
| The 19 conjugacy class representatives for $C_4:D_4.D_4$ |
| Character table for $C_4:D_4.D_4$ |
Intermediate fields
| \(\Q(\sqrt{53}) \), 4.2.30899.1, 8.0.6122800213013.1 |
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 | $16$ | ${\href{/LocalNumberField/3.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ | $16$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ | $16$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | R | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $11$ | $\Q_{11}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{11}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.4.3.2 | $x^{4} - 11$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
| 11.8.7.1 | $x^{8} + 33$ | $8$ | $1$ | $7$ | $QD_{16}$ | $[\ ]_{8}^{2}$ | |
| 53 | Data not computed | ||||||