Normalized defining polynomial
\( x^{16} - 5 x^{15} + 15 x^{14} - 32 x^{13} + 56 x^{12} - 85 x^{11} + 114 x^{10} - 134 x^{9} + 131 x^{8} - 98 x^{7} + 42 x^{6} + 9 x^{5} - 28 x^{4} + 18 x^{3} - x^{2} - 3 x + 1 \)
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: | \(6561000000000000\)\(\medspace = 2^{12}\cdot 3^{8}\cdot 5^{12}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $9.74$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{13129} a^{15} - \frac{1370}{13129} a^{14} + \frac{5747}{13129} a^{13} + \frac{6455}{13129} a^{12} - \frac{1460}{13129} a^{11} - \frac{147}{691} a^{10} + \frac{271}{691} a^{9} - \frac{4504}{13129} a^{8} + \frac{3719}{13129} a^{7} + \frac{4390}{13129} a^{6} - \frac{5484}{13129} a^{5} + \frac{2139}{13129} a^{4} - \frac{5125}{13129} a^{3} - \frac{2114}{13129} a^{2} - \frac{2771}{13129} a + \frac{1260}{13129}$
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: | \( \frac{4711}{13129} a^{15} - \frac{20860}{13129} a^{14} + \frac{67764}{13129} a^{13} - \frac{154807}{13129} a^{12} + \frac{290374}{13129} a^{11} - \frac{24320}{691} a^{10} + \frac{34263}{691} a^{9} - \frac{815878}{13129} a^{8} + \frac{872637}{13129} a^{7} - \frac{771496}{13129} a^{6} + \frac{501650}{13129} a^{5} - \frac{163791}{13129} a^{4} - \frac{65289}{13129} a^{3} + \frac{110889}{13129} a^{2} - \frac{30213}{13129} a - \frac{11577}{13129} \) (order $30$) | 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: | \( 175.01473438 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
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Class number formula
Galois group
$C_4\times D_4$ (as 16T19):
A solvable group of order 32 |
The 20 conjugacy class representatives for $C_4 \times D_4$ |
Character table for $C_4 \times D_4$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | Deg 32 |
Degree 16 siblings: | 16.0.331776000000000000.1, 16.8.26873856000000000000.4, 16.0.26873856000000000000.11 |
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 | ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
$2$ | 2.4.0.1 | $x^{4} - x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
2.4.0.1 | $x^{4} - x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
2.8.12.1 | $x^{8} + 6 x^{6} + 8 x^{5} + 16$ | $2$ | $4$ | $12$ | $C_4\times C_2$ | $[3]^{4}$ | |
$3$ | 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
5 | Data not computed |