Normalized defining polynomial
\( x^{16} - 6 x^{15} + 28 x^{14} - 96 x^{13} + 266 x^{12} - 468 x^{11} + 820 x^{10} - 260 x^{9} - 1004 x^{8} + 2460 x^{7} - 960 x^{6} - 136 x^{5} + 604 x^{4} + 104 x^{3} + 1440 x^{2} - 1768 x + 716 \)
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: | \(806950076416000000000000=2^{28}\cdot 5^{12}\cdot 11^{4}\cdot 29^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $31.20$ | 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: | $2, 5, 11, 29$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{4} a^{12} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{13} - \frac{1}{2} a^{5}$, $\frac{1}{44} a^{14} + \frac{1}{44} a^{13} - \frac{1}{22} a^{12} - \frac{1}{11} a^{11} + \frac{1}{11} a^{10} - \frac{3}{22} a^{9} - \frac{2}{11} a^{8} + \frac{4}{11} a^{7} - \frac{1}{22} a^{6} + \frac{3}{22} a^{5} - \frac{2}{11} a^{4} + \frac{1}{11} a^{3} + \frac{1}{11} a^{2} - \frac{4}{11} a - \frac{2}{11}$, $\frac{1}{42479774329623642733002644} a^{15} - \frac{59465582896431198237302}{10619943582405910683250661} a^{14} - \frac{100038457367018150801438}{965449416582355516659151} a^{13} + \frac{680339995687066314459682}{10619943582405910683250661} a^{12} - \frac{2384708837163229083284131}{21239887164811821366501322} a^{11} - \frac{2417009137043409875769021}{21239887164811821366501322} a^{10} + \frac{286553948363304699245328}{10619943582405910683250661} a^{9} - \frac{33660897826854215569491}{1930898833164711033318302} a^{8} + \frac{4901759625860955215283303}{21239887164811821366501322} a^{7} + \frac{952739359918539655212119}{10619943582405910683250661} a^{6} + \frac{4604842336634151166375151}{10619943582405910683250661} a^{5} - \frac{1888146441872868969363377}{10619943582405910683250661} a^{4} - \frac{2684500524606266821070360}{10619943582405910683250661} a^{3} + \frac{3906649051339918492701266}{10619943582405910683250661} a^{2} - \frac{2727491715327471451980818}{10619943582405910683250661} a + \frac{3553227551717327440221239}{10619943582405910683250661}$
Class group and class number
$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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: | \( 28495.2195483 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^4.C_2^3.C_2$ (as 16T547):
| A solvable group of order 256 |
| The 40 conjugacy class representatives for $C_2^4.C_2^3.C_2$ |
| Character table for $C_2^4.C_2^3.C_2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{20})^+\), 4.4.4400.1, 4.4.22000.1, 8.0.224576000000.3, 8.0.8983040000.1, 8.8.7744000000.1 |
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 | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| $5$ | 5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
| 5.4.3.2 | $x^{4} - 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $11$ | 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 11.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 11.4.2.1 | $x^{4} + 143 x^{2} + 5929$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 11.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| $29$ | 29.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 29.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 29.4.2.1 | $x^{4} + 145 x^{2} + 7569$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 29.4.0.1 | $x^{4} - x + 19$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |