Normalized defining polynomial
\( x^{16} - 4 x^{15} + 9 x^{14} - 14 x^{13} + 16 x^{12} - 9 x^{11} - 8 x^{10} + 27 x^{9} - 35 x^{8} + 27 x^{7} - 8 x^{6} - 9 x^{5} + 16 x^{4} - 14 x^{3} + 9 x^{2} - 4 x + 1 \)
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: | \(6468474822995241\)\(\medspace = 3^{2}\cdot 7^{2}\cdot 19^{4}\cdot 103^{4}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $9.73$ | 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: | $3, 7, 19, 103$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$
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: | \( -1 \) (order $2$) | 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: | \( 11.6239832087 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
A solvable group of order 6144 |
The 105 conjugacy class representatives for t16n1664 are not computed |
Character table for t16n1664 is not computed |
Intermediate fields
4.4.1957.1, 8.4.80426829.1, 8.2.11489547.1, 8.2.26808943.1 |
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 | ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
$3$ | 3.4.2.1 | $x^{4} + 9 x^{2} + 36$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
3.6.0.1 | $x^{6} - x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
3.6.0.1 | $x^{6} - x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
$7$ | 7.4.2.1 | $x^{4} + 35 x^{2} + 441$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
7.6.0.1 | $x^{6} + 3 x^{2} - x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
7.6.0.1 | $x^{6} + 3 x^{2} - x + 5$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
$19$ | 19.4.0.1 | $x^{4} - 2 x + 10$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
19.4.0.1 | $x^{4} - 2 x + 10$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
19.4.2.1 | $x^{4} + 57 x^{2} + 1444$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
$103$ | 103.4.2.1 | $x^{4} + 927 x^{2} + 265225$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
103.4.0.1 | $x^{4} - x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
103.4.0.1 | $x^{4} - x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
103.4.2.1 | $x^{4} + 927 x^{2} + 265225$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |