Normalized defining polynomial
\( x^{8} - 8x^{6} + 12x^{4} - 24x^{3} - 28x^{2} + 24x + 31 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(1212153856\) \(\medspace = 2^{22}\cdot 17^{2}\) | 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: | \(13.66\) | 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: | $2^{11/4}17^{1/2}\approx 27.736837922354518$ | ||
Ramified primes: | \(2\), \(17\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{632}a^{7}+\frac{7}{632}a^{6}-\frac{117}{632}a^{5}+\frac{129}{632}a^{4}+\frac{125}{632}a^{3}+\frac{219}{632}a^{2}+\frac{83}{632}a+\frac{289}{632}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | 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: | $\frac{7}{316}a^{7}-\frac{15}{158}a^{6}-\frac{29}{316}a^{5}+\frac{48}{79}a^{4}-\frac{73}{316}a^{3}-\frac{63}{158}a^{2}+\frac{423}{316}a+\frac{91}{79}$, $\frac{67}{632}a^{7}-\frac{163}{632}a^{6}-\frac{255}{632}a^{5}+\frac{743}{632}a^{4}-\frac{473}{632}a^{3}-\frac{1127}{632}a^{2}+\frac{1137}{632}a+\frac{1983}{632}$, $\frac{17}{632}a^{7}-\frac{39}{632}a^{6}-\frac{93}{632}a^{5}+\frac{139}{632}a^{4}+\frac{229}{632}a^{3}-\frac{227}{632}a^{2}-\frac{485}{632}a+\frac{1595}{632}$, $\frac{29}{632}a^{7}-\frac{113}{632}a^{6}+\frac{83}{632}a^{5}+\frac{265}{632}a^{4}-\frac{799}{632}a^{3}+\frac{347}{632}a^{2}+\frac{195}{632}a-\frac{783}{632}$, $\frac{75}{632}a^{7}+\frac{51}{632}a^{6}-\frac{559}{632}a^{5}-\frac{279}{632}a^{4}+\frac{527}{632}a^{3}-\frac{1745}{632}a^{2}-\frac{2623}{632}a-\frac{919}{632}$ | 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: | \( 85.5081560743 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 85.5081560743 \cdot 1}{2\cdot\sqrt{1212153856}}\cr\approx \mathstrut & 0.775672494044 \end{aligned}\]
Galois group
$C_2^3:C_4$ (as 8T20):
A solvable group of order 32 |
The 11 conjugacy class representatives for $C_2^3: C_4$ |
Character table for $C_2^3: C_4$ |
Intermediate fields
\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Degree 8 siblings: | data not computed |
Degree 16 siblings: | 16.8.122718822702791268499456.3, 16.0.119842600295694598144.8, 16.0.424632604507928264704.1, some data not computed |
Minimal sibling: | 8.0.1368408064.3 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.11.1 | $x^{4} + 8 x^{3} + 4 x^{2} + 2$ | $4$ | $1$ | $11$ | $C_4$ | $[3, 4]$ |
2.4.11.1 | $x^{4} + 8 x^{3} + 4 x^{2} + 2$ | $4$ | $1$ | $11$ | $C_4$ | $[3, 4]$ | |
\(17\) | $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
17.2.1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |