Normalized defining polynomial
\( x^{16} - 4 x^{15} + 32 x^{14} - 92 x^{13} + 362 x^{12} - 764 x^{11} + 1772 x^{10} - 2606 x^{9} + 3751 x^{8} - 3676 x^{7} + 2818 x^{6} - 830 x^{5} + 579 x^{4} - 394 x^{3} + 211 x^{2} + 270 x + 89 \)
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: | \(3333974497696153600000000=2^{24}\cdot 5^{8}\cdot 37^{4}\cdot 521^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $34.09$ | 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: | $2, 5, 37, 521$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{41} a^{13} + \frac{14}{41} a^{12} - \frac{5}{41} a^{11} + \frac{14}{41} a^{10} - \frac{12}{41} a^{9} + \frac{4}{41} a^{8} + \frac{2}{41} a^{7} - \frac{18}{41} a^{6} + \frac{14}{41} a^{5} + \frac{12}{41} a^{4} - \frac{1}{41} a^{3} + \frac{9}{41} a^{2} - \frac{13}{41} a - \frac{9}{41}$, $\frac{1}{4551} a^{14} + \frac{34}{4551} a^{13} + \frac{1259}{4551} a^{12} - \frac{138}{1517} a^{11} + \frac{1252}{4551} a^{10} - \frac{475}{1517} a^{9} + \frac{9}{37} a^{8} + \frac{883}{4551} a^{7} - \frac{6}{1517} a^{6} + \frac{357}{1517} a^{5} - \frac{1934}{4551} a^{4} + \frac{10}{1517} a^{3} - \frac{776}{4551} a^{2} + \frac{469}{4551} a + \frac{763}{4551}$, $\frac{1}{67154431433379142503} a^{15} + \frac{374135689849058}{67154431433379142503} a^{14} + \frac{137918837204396309}{22384810477793047501} a^{13} + \frac{2474045550854859233}{9593490204768448929} a^{12} - \frac{11102037846785412914}{67154431433379142503} a^{11} + \frac{2286186149656978468}{67154431433379142503} a^{10} + \frac{8190132139258530031}{22384810477793047501} a^{9} - \frac{26518689716934429242}{67154431433379142503} a^{8} + \frac{25755157068042773128}{67154431433379142503} a^{7} + \frac{5117556728555768964}{22384810477793047501} a^{6} - \frac{10611098455921261538}{67154431433379142503} a^{5} + \frac{11508896117688612427}{67154431433379142503} a^{4} - \frac{12620137887251150747}{67154431433379142503} a^{3} - \frac{6286011832174058533}{67154431433379142503} a^{2} + \frac{13144383709322855939}{67154431433379142503} a + \frac{11624223902119721041}{67154431433379142503}$
Class group and class number
Trivial group, which has order $1$ (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: | \( 283101.129435 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^6.C_2^2$ (as 16T528):
| A solvable group of order 256 |
| The 46 conjugacy class representatives for $C_2^6.C_2^2$ |
| Character table for $C_2^6.C_2^2$ is not computed |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{2}, \sqrt{5})\), 8.0.1333760000.1, 8.0.3504640000.1, 8.8.1825917440000.1 |
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 | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ | R | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.8.12.1 | $x^{8} + 6 x^{6} + 8 x^{5} + 16$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ |
| 2.8.12.1 | $x^{8} + 6 x^{6} + 8 x^{5} + 16$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
| $5$ | 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 5.8.4.1 | $x^{8} + 10 x^{6} + 125 x^{4} + 2500$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $37$ | 37.4.2.2 | $x^{4} - 37 x^{2} + 6845$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
| 37.4.2.2 | $x^{4} - 37 x^{2} + 6845$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
| 37.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 37.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
| 521 | Data not computed | ||||||