Normalized defining polynomial
\( x^{16} - 4 x^{14} - 12 x^{13} + 6 x^{12} + 56 x^{11} + 48 x^{10} - 92 x^{9} - 165 x^{8} - 192 x^{7} + 692 x^{6} - 388 x^{5} + 816 x^{4} - 800 x^{3} + 668 x^{2} - 344 x + 71 \)
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: | \(760879655786905600000000=2^{40}\cdot 5^{8}\cdot 11^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $31.09$ | 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: | $2, 5, 11$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{11} a^{12} + \frac{3}{11} a^{11} + \frac{3}{11} a^{10} + \frac{3}{11} a^{9} - \frac{4}{11} a^{8} + \frac{4}{11} a^{7} + \frac{1}{11} a^{6} + \frac{5}{11} a^{5} + \frac{4}{11} a^{4} + \frac{2}{11} a^{2} + \frac{4}{11} a + \frac{1}{11}$, $\frac{1}{55} a^{13} - \frac{2}{55} a^{12} - \frac{1}{55} a^{11} + \frac{2}{11} a^{10} - \frac{19}{55} a^{9} - \frac{9}{55} a^{8} + \frac{5}{11} a^{7} + \frac{23}{55} a^{5} + \frac{2}{55} a^{4} - \frac{4}{11} a^{3} - \frac{6}{55} a^{2} + \frac{14}{55} a - \frac{27}{55}$, $\frac{1}{605} a^{14} - \frac{4}{605} a^{13} + \frac{8}{605} a^{12} - \frac{28}{605} a^{11} + \frac{86}{605} a^{10} - \frac{16}{55} a^{9} - \frac{197}{605} a^{8} + \frac{38}{121} a^{7} - \frac{137}{605} a^{6} + \frac{201}{605} a^{5} - \frac{169}{605} a^{4} + \frac{89}{605} a^{3} - \frac{74}{605} a^{2} - \frac{51}{121} a + \frac{169}{605}$, $\frac{1}{4609804122115225} a^{15} - \frac{247878561307}{419073102010475} a^{14} - \frac{5915225533476}{921960824423045} a^{13} - \frac{4006224001952}{4609804122115225} a^{12} - \frac{402229254410393}{921960824423045} a^{11} + \frac{1802860751401471}{4609804122115225} a^{10} - \frac{1799760221340124}{4609804122115225} a^{9} - \frac{1507612321172434}{4609804122115225} a^{8} - \frac{426640990509782}{4609804122115225} a^{7} + \frac{1203141163882597}{4609804122115225} a^{6} - \frac{712415542255717}{4609804122115225} a^{5} + \frac{187565414091381}{4609804122115225} a^{4} - \frac{633188776180866}{4609804122115225} a^{3} + \frac{953842865299312}{4609804122115225} a^{2} - \frac{1089518432239791}{4609804122115225} a + \frac{815049460310783}{4609804122115225}$
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: | \( 328916.376179 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^2.C_2^5.C_2$ (as 16T486):
| A solvable group of order 256 |
| The 34 conjugacy class representatives for $C_2^2.C_2^5.C_2$ |
| Character table for $C_2^2.C_2^5.C_2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.2.1600.1, 4.0.4400.1, 4.2.17600.2, 8.0.4956160000.4 |
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 | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$ |
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 | Data not computed | ||||||
| $5$ | 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.2.1.1 | $x^{2} - 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $11$ | $\Q_{11}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| $\Q_{11}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
| 11.2.1.2 | $x^{2} + 33$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 11.2.1.1 | $x^{2} - 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |