Normalized defining polynomial
\( x^{16} - 4 x^{15} + 11 x^{14} - 24 x^{13} + 46 x^{12} - 75 x^{11} + 104 x^{10} - 121 x^{9} + 121 x^{8} - 107 x^{7} + 86 x^{6} - 63 x^{5} + 43 x^{4} - 27 x^{3} + 14 x^{2} - 5 x + 1 \)
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: | \(5605632412826517\)\(\medspace = 3^{10}\cdot 37^{7}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $9.64$ | 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: | $3, 37$ | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
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$|\Aut(K/\Q)|$: | $4$ | ||
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} + \frac{2}{13} a^{13} + \frac{6}{13} a^{12} + \frac{4}{13} a^{11} - \frac{6}{13} a^{10} + \frac{3}{13} a^{9} + \frac{3}{13} a^{8} + \frac{2}{13} a^{7} + \frac{4}{13} a^{6} + \frac{5}{13} a^{4} + \frac{6}{13} a^{3} - \frac{6}{13} a^{2} + \frac{4}{13} a - \frac{3}{13}$, $\frac{1}{13} a^{15} + \frac{2}{13} a^{13} + \frac{5}{13} a^{12} - \frac{1}{13} a^{11} + \frac{2}{13} a^{10} - \frac{3}{13} a^{9} - \frac{4}{13} a^{8} + \frac{5}{13} a^{6} + \frac{5}{13} a^{5} - \frac{4}{13} a^{4} - \frac{5}{13} a^{3} + \frac{3}{13} a^{2} + \frac{2}{13} a + \frac{6}{13}$
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: | \( \frac{14}{13} a^{15} - \frac{56}{13} a^{14} + \frac{150}{13} a^{13} - \frac{331}{13} a^{12} + \frac{620}{13} a^{11} - 78 a^{10} + \frac{1376}{13} a^{9} - \frac{1589}{13} a^{8} + \frac{1539}{13} a^{7} - \frac{1337}{13} a^{6} + \frac{1032}{13} a^{5} - \frac{752}{13} a^{4} + \frac{491}{13} a^{3} - \frac{298}{13} a^{2} + \frac{142}{13} a - \frac{34}{13} \) (order $6$) | 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: | \( 32.1900879299 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
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Class number formula
Galois group
$(C_2^3\times C_4).D_4$ (as 16T675):
A solvable group of order 256 |
The 31 conjugacy class representatives for $(C_2^3\times C_4).D_4$ |
Character table for $(C_2^3\times C_4).D_4$ is not computed |
Intermediate fields
\(\Q(\sqrt{-3}) \), 4.0.333.1, 8.0.4102893.1 |
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 | $16$ | R | $16$ | ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ | $16$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ | $16$ | $16$ | ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{6}$ | R | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ | $16$ |
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 |
---|---|---|---|---|---|---|---|
$3$ | 3.8.6.1 | $x^{8} + 9 x^{4} + 36$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
$37$ | 37.4.3.4 | $x^{4} + 296$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
37.4.0.1 | $x^{4} - x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
37.8.4.1 | $x^{8} + 5476 x^{4} - 50653 x^{2} + 7496644$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |