Normalized defining polynomial
\( x^{16} - 4 x^{15} + 6 x^{14} - 14 x^{12} + 24 x^{11} - 18 x^{10} + 15 x^{8} - 20 x^{7} + 22 x^{6} - 24 x^{5} + 24 x^{4} - 20 x^{3} + 12 x^{2} - 4 x + 1 \)
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: | \(8084777718513664\)\(\medspace = 2^{36}\cdot 7^{6}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $9.87$ | 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: | $2, 7$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5549} a^{15} + \frac{2615}{5549} a^{14} + \frac{1225}{5549} a^{13} + \frac{953}{5549} a^{12} - \frac{1157}{5549} a^{11} - \frac{405}{5549} a^{10} - \frac{854}{5549} a^{9} - \frac{379}{5549} a^{8} + \frac{685}{5549} a^{7} + \frac{1668}{5549} a^{6} + \frac{1451}{5549} a^{5} - \frac{920}{5549} a^{4} - \frac{1190}{5549} a^{3} + \frac{1908}{5549} a^{2} - \frac{2585}{5549} a - \frac{339}{5549}$
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: | \( -\frac{6779}{5549} a^{15} + \frac{24166}{5549} a^{14} - \frac{30716}{5549} a^{13} - \frac{12449}{5549} a^{12} + \frac{91350}{5549} a^{11} - \frac{128887}{5549} a^{10} + \frac{68247}{5549} a^{9} + \frac{38897}{5549} a^{8} - \frac{98984}{5549} a^{7} + \frac{95823}{5549} a^{6} - \frac{97834}{5549} a^{5} + \frac{110584}{5549} a^{4} - \frac{112216}{5549} a^{3} + \frac{83622}{5549} a^{2} - \frac{38870}{5549} a + \frac{6344}{5549} \) (order $8$) | 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: | \( 60.0447869363 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
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Class number formula
Galois group
A solvable group of order 64 |
The 16 conjugacy class representatives for $D_4:D_4$ |
Character table for $D_4:D_4$ |
Intermediate fields
\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), 4.2.1792.1, 4.2.448.1, \(\Q(\zeta_{8})\), 8.2.89915392.1 x2, 8.2.22478848.1 x2, 8.0.3211264.1 |
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 |
Degree 16 siblings: | data not computed |
Degree 32 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 | R | ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ | R | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$ |
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 | ||||||
$7$ | 7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.2.0.1 | $x^{2} - x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
7.8.6.2 | $x^{8} - 49 x^{4} + 3969$ | $4$ | $2$ | $6$ | $D_4$ | $[\ ]_{4}^{2}$ |