Normalized defining polynomial
\( x^{16} - 8x^{12} + 13x^{8} + 12x^{4} + 4 \)
Invariants
Degree: | $16$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(405661806804141604864\) \(\medspace = 2^{46}\cdot 7^{8}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(19.41\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
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Galois root discriminant: | $2^{3}7^{1/2}\approx 21.166010488516726$ | ||
Ramified primes: | \(2\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{8}a^{12}-\frac{1}{2}a^{10}+\frac{1}{4}a^{8}+\frac{1}{8}a^{4}-\frac{1}{2}a^{2}-\frac{1}{4}$, $\frac{1}{8}a^{13}-\frac{1}{2}a^{11}+\frac{1}{4}a^{9}+\frac{1}{8}a^{5}-\frac{1}{2}a^{3}-\frac{1}{4}a$, $\frac{1}{16}a^{14}+\frac{1}{8}a^{10}-\frac{1}{2}a^{8}+\frac{1}{16}a^{6}+\frac{3}{8}a^{2}-\frac{1}{2}$, $\frac{1}{16}a^{15}+\frac{1}{8}a^{11}-\frac{1}{2}a^{9}+\frac{1}{16}a^{7}+\frac{3}{8}a^{3}-\frac{1}{2}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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);
oscar: torsion_units_generator(OK)
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Fundamental units: | $\frac{3}{16}a^{14}-\frac{13}{8}a^{10}+\frac{1}{2}a^{8}+\frac{51}{16}a^{6}-2a^{4}+\frac{9}{8}a^{2}-\frac{1}{2}$, $\frac{1}{8}a^{14}-\frac{3}{4}a^{10}+\frac{9}{8}a^{6}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{14}-\frac{1}{4}a^{12}-\frac{3}{4}a^{10}+\frac{3}{2}a^{8}+\frac{9}{8}a^{6}-\frac{5}{4}a^{4}-\frac{5}{4}a^{2}-\frac{3}{2}$, $\frac{3}{16}a^{15}-\frac{1}{16}a^{14}-\frac{1}{8}a^{13}-\frac{9}{8}a^{11}-\frac{1}{8}a^{10}+\frac{5}{4}a^{9}-\frac{1}{2}a^{8}+\frac{19}{16}a^{7}+\frac{31}{16}a^{6}-\frac{25}{8}a^{5}+2a^{4}+\frac{5}{8}a^{3}-\frac{3}{8}a^{2}-\frac{1}{4}a+\frac{1}{2}$, $\frac{3}{16}a^{15}+\frac{3}{16}a^{14}-\frac{3}{8}a^{12}-\frac{13}{8}a^{11}-\frac{9}{8}a^{10}+\frac{1}{2}a^{9}+\frac{7}{4}a^{8}+\frac{51}{16}a^{7}+\frac{19}{16}a^{6}-2a^{5}-\frac{3}{8}a^{4}+\frac{9}{8}a^{3}+\frac{5}{8}a^{2}+\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{4}a^{15}-\frac{1}{8}a^{14}-\frac{3}{8}a^{13}-2a^{11}+\frac{3}{4}a^{10}+\frac{9}{4}a^{9}+a^{8}+\frac{17}{4}a^{7}-\frac{1}{8}a^{6}-\frac{19}{8}a^{5}-4a^{4}-a^{3}-\frac{11}{4}a^{2}-\frac{9}{4}a-2$, $\frac{5}{16}a^{15}-\frac{1}{16}a^{14}-\frac{1}{8}a^{13}+\frac{1}{8}a^{12}-\frac{23}{8}a^{11}+\frac{3}{8}a^{10}+\frac{5}{4}a^{9}-\frac{5}{4}a^{8}+\frac{101}{16}a^{7}-\frac{1}{16}a^{6}-\frac{25}{8}a^{5}+\frac{25}{8}a^{4}+\frac{11}{8}a^{3}-\frac{15}{8}a^{2}-\frac{1}{4}a+\frac{1}{4}$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 10856.0619541 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{8}\cdot 10856.0619541 \cdot 1}{2\cdot\sqrt{405661806804141604864}}\cr\approx \mathstrut & 0.654634974201 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 32 |
The 11 conjugacy class representatives for $D_8:C_2$ |
Character table for $D_8:C_2$ |
Intermediate fields
\(\Q(\sqrt{-7}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{-2}) \), 4.0.1568.1, 4.0.392.1, \(\Q(\sqrt{-2}, \sqrt{-7})\), 8.0.39337984.3 |
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 | ${\href{/padicField/3.8.0.1}{8} }^{2}$ | ${\href{/padicField/5.8.0.1}{8} }^{2}$ | R | ${\href{/padicField/11.2.0.1}{2} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }^{4}$ | ${\href{/padicField/13.8.0.1}{8} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{8}$ | ${\href{/padicField/19.8.0.1}{8} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}$ | ${\href{/padicField/31.2.0.1}{2} }^{8}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}$ | ${\href{/padicField/59.8.0.1}{8} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.22.10 | $x^{8} - 8 x^{7} + 88 x^{6} - 96 x^{5} + 204 x^{4} - 240 x^{3} + 720 x^{2} + 900$ | $4$ | $2$ | $22$ | $Q_8:C_2$ | $[2, 3, 4]^{2}$ |
2.8.24.29 | $x^{8} + 8 x^{7} + 14 x^{4} + 12 x^{2} + 8 x + 6$ | $8$ | $1$ | $24$ | $Q_8:C_2$ | $[2, 3, 4]^{2}$ | |
\(7\) | 7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |