Normalized defining polynomial
\( x^{16} - 2 x^{15} + 2 x^{14} - 3 x^{13} + 11 x^{12} - 19 x^{11} + 10 x^{10} + 21 x^{9} - 44 x^{8} + 21 x^{7} + 40 x^{6} - 89 x^{5} + 92 x^{4} - 60 x^{3} + 26 x^{2} - 7 x + 1 \)
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: | \(7058653305387264\)\(\medspace = 2^{8}\cdot 3^{14}\cdot 7^{8}\) | sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
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Root discriminant: | $9.78$ | 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: | $2, 3, 7$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{40937} a^{15} - \frac{197}{3149} a^{14} + \frac{3681}{40937} a^{13} - \frac{4172}{40937} a^{12} - \frac{646}{3149} a^{11} - \frac{1462}{40937} a^{10} + \frac{16001}{40937} a^{9} - \frac{9538}{40937} a^{8} + \frac{138}{611} a^{7} + \frac{1093}{40937} a^{6} - \frac{13231}{40937} a^{5} + \frac{3141}{40937} a^{4} - \frac{14075}{40937} a^{3} - \frac{515}{3149} a^{2} - \frac{1544}{3149} a - \frac{11694}{40937}$
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: | \( -\frac{324}{3149} a^{15} - \frac{4721}{3149} a^{14} + \frac{3976}{3149} a^{13} + \frac{807}{3149} a^{12} + \frac{6514}{3149} a^{11} - \frac{39599}{3149} a^{10} + \frac{36718}{3149} a^{9} + \frac{32633}{3149} a^{8} - \frac{1566}{47} a^{7} + \frac{67834}{3149} a^{6} + \frac{86078}{3149} a^{5} - \frac{180050}{3149} a^{4} + \frac{139104}{3149} a^{3} - \frac{54014}{3149} a^{2} + \frac{3792}{3149} a + \frac{3758}{3149} \) (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: | \( 37.0149239843 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
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Class number formula
Galois group
A solvable group of order 32 |
The 14 conjugacy class representatives for $D_8:C_2$ |
Character table for $D_8:C_2$ |
Intermediate fields
\(\Q(\sqrt{21}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-7}) \), 4.2.1323.1 x2, 4.0.189.1 x2, \(\Q(\sqrt{-3}, \sqrt{-7})\), 8.0.1750329.1 |
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 | ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ | R | ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ | ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
$2$ | 2.8.8.4 | $x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$ | $2$ | $4$ | $8$ | $C_8$ | $[2]^{4}$ |
2.8.0.1 | $x^{8} + x^{4} + x^{3} + x + 1$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | |
3 | Data not computed | ||||||
$7$ | 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |