Normalized defining polynomial
\( x^{16} - 4 x^{15} + 8 x^{14} - 6 x^{13} - 10 x^{12} + 28 x^{11} - 8 x^{10} - 90 x^{9} + 253 x^{8} - 396 x^{7} + 433 x^{6} - 352 x^{5} + 218 x^{4} - 102 x^{3} + 35 x^{2} - 8 x + 1 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
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Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
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Discriminant: | \(4897760256000000\)\(\medspace = 2^{16}\cdot 3^{14}\cdot 5^{6}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $9.56$ | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
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Ramified primes: | $2, 3, 5$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
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$|\Aut(K/\Q)|$: | $8$ | ||
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{31} a^{15} - \frac{1}{31} a^{14} + \frac{5}{31} a^{13} + \frac{9}{31} a^{12} - \frac{14}{31} a^{11} - \frac{14}{31} a^{10} + \frac{12}{31} a^{9} + \frac{8}{31} a^{8} - \frac{2}{31} a^{7} + \frac{1}{31} a^{6} + \frac{2}{31} a^{5} - \frac{5}{31} a^{4} - \frac{14}{31} a^{3} + \frac{11}{31} a^{2} + \frac{6}{31} a + \frac{10}{31}$
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
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Torsion generator: | \( \frac{628}{31} a^{15} - \frac{2705}{31} a^{14} + \frac{5248}{31} a^{13} - \frac{3555}{31} a^{12} - \frac{7893}{31} a^{11} + \frac{19945}{31} a^{10} - \frac{3624}{31} a^{9} - \frac{64912}{31} a^{8} + \frac{170050}{31} a^{7} - \frac{249666}{31} a^{6} + \frac{250775}{31} a^{5} - \frac{183529}{31} a^{4} + \frac{99119}{31} a^{3} - \frac{38848}{31} a^{2} + \frac{10030}{31} a - \frac{1408}{31} \) (order $12$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
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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 | sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
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Regulator: | \( 59.5214550029 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
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Class number formula
Galois group
$(C_4\times C_8):C_2$ (as 16T114):
A solvable group of order 64 |
The 28 conjugacy class representatives for $(C_4\times C_8):C_2$ |
Character table for $(C_4\times C_8):C_2$ is not computed |
Intermediate fields
\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{12})\), 8.0.4665600.1 |
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 | R | R | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 | ||||||
3 | Data not computed | ||||||
$5$ | 5.8.0.1 | $x^{8} + x^{2} - 2 x + 3$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |
5.8.6.4 | $x^{8} - 5 x^{4} + 50$ | $4$ | $2$ | $6$ | $C_8$ | $[\ ]_{4}^{2}$ |