Normalized defining polynomial
\( x^{16} + 4 x^{14} - 12 x^{12} - 1813 x^{10} + 15464 x^{8} - 29008 x^{6} - 3072 x^{4} + 16384 x^{2} + 65536 \)
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: | \(8813863398580541437890625=5^{8}\cdot 41^{12}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $36.23$ | 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: | $5, 41$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois over $\Q$. | |||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{24} a^{8} - \frac{1}{4} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{4} + \frac{11}{24} a^{2} + \frac{1}{4} a - \frac{1}{3}$, $\frac{1}{48} a^{9} + \frac{1}{6} a^{7} + \frac{1}{12} a^{5} + \frac{11}{48} a^{3} + \frac{1}{12} a$, $\frac{1}{192} a^{10} - \frac{1}{4} a^{7} - \frac{1}{16} a^{6} + \frac{9}{64} a^{4} + \frac{5}{16} a^{2} + \frac{1}{4} a + \frac{1}{3}$, $\frac{1}{768} a^{11} - \frac{1}{96} a^{9} - \frac{43}{192} a^{7} - \frac{5}{768} a^{5} + \frac{41}{192} a^{3} - \frac{1}{3} a$, $\frac{1}{9200640} a^{12} - \frac{1}{1536} a^{11} - \frac{287}{383360} a^{10} + \frac{1}{192} a^{9} + \frac{6599}{766720} a^{8} - \frac{53}{384} a^{7} - \frac{300247}{3066880} a^{6} + \frac{5}{1536} a^{5} + \frac{180459}{766720} a^{4} - \frac{41}{384} a^{3} + \frac{1199}{71880} a^{2} - \frac{1}{12} a + \frac{1001}{5990}$, $\frac{1}{18401280} a^{13} + \frac{1273}{4600320} a^{11} - \frac{4163}{4600320} a^{9} - \frac{2961301}{18401280} a^{7} + \frac{263201}{2300160} a^{5} - \frac{51711}{383360} a^{3} - \frac{2987}{35940} a - \frac{1}{2}$, $\frac{1}{3165020160} a^{14} - \frac{1}{18401280} a^{12} - \frac{98797}{52750336} a^{10} - \frac{1}{96} a^{9} + \frac{14156587}{3165020160} a^{8} + \frac{1}{6} a^{7} - \frac{9307781}{395627520} a^{6} - \frac{1}{24} a^{5} - \frac{2681309}{39562752} a^{4} + \frac{37}{96} a^{3} + \frac{26299}{71880} a^{2} + \frac{5}{24} a + \frac{141128}{386355}$, $\frac{1}{12660080640} a^{15} - \frac{1}{73605120} a^{13} + \frac{115721}{633004032} a^{11} - \frac{1}{384} a^{10} + \frac{26698169}{4220026880} a^{9} - \frac{1}{48} a^{8} + \frac{47724553}{527503360} a^{7} + \frac{11}{96} a^{6} + \frac{3397343}{158251008} a^{5} - \frac{59}{384} a^{4} + \frac{216011}{1150080} a^{3} - \frac{37}{96} a^{2} - \frac{315791}{772710} a - \frac{1}{2}$
Class group and class number
$C_{3}\times C_{3}$, which has order $9$ (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: | \( 525465.124463 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^2:C_4$ (as 16T10):
| A solvable group of order 16 |
| The 10 conjugacy class representatives for $C_2^2 : C_4$ |
| Character table for $C_2^2 : C_4$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 8 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 | ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ | R | ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 5.4.2.1 | $x^{4} + 15 x^{2} + 100$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| $41$ | 41.8.6.1 | $x^{8} - 9881 x^{4} + 34857216$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 41.8.6.1 | $x^{8} - 9881 x^{4} + 34857216$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |