Normalized defining polynomial
\( x^{8} + 4x^{6} - 8x^{5} + 6x^{4} + 8x^{3} - 36x^{2} - 24x + 2 \)
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: | \(4848615424\) \(\medspace = 2^{24}\cdot 17^{2}\) | 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: | \(16.24\) | 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: | $2^{3}17^{1/2}\approx 32.984845004941285$ | ||
Ramified primes: | \(2\), \(17\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{10481}a^{7}-\frac{4743}{10481}a^{6}+\frac{3827}{10481}a^{5}+\frac{1623}{10481}a^{4}-\frac{4829}{10481}a^{3}+\frac{2970}{10481}a^{2}-\frac{6}{223}a-\frac{4066}{10481}$
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: | $\frac{148}{10481}a^{7}+\frac{263}{10481}a^{6}+\frac{422}{10481}a^{5}-\frac{859}{10481}a^{4}-\frac{1984}{10481}a^{3}-\frac{642}{10481}a^{2}+\frac{4}{223}a-\frac{25313}{10481}$, $\frac{4478}{10481}a^{7}-\frac{4648}{10481}a^{6}+\frac{21833}{10481}a^{5}-\frac{58425}{10481}a^{4}+\frac{81889}{10481}a^{3}-\frac{42653}{10481}a^{2}-\frac{2784}{223}a+\frac{39873}{10481}$, $\frac{1755}{10481}a^{7}-\frac{2051}{10481}a^{6}+\frac{8545}{10481}a^{5}-\frac{23429}{10481}a^{4}+\frac{35677}{10481}a^{3}-\frac{17669}{10481}a^{2}-\frac{1164}{223}a+\frac{22693}{10481}$, $\frac{4363}{10481}a^{7}-\frac{4215}{10481}a^{6}+\frac{21930}{10481}a^{5}-\frac{56412}{10481}a^{4}+\frac{81731}{10481}a^{3}-\frac{48811}{10481}a^{2}-\frac{2317}{223}a+\frac{4375}{10481}$, $\frac{16521}{10481}a^{7}-\frac{13628}{10481}a^{6}+\frac{77842}{10481}a^{5}-\frac{195954}{10481}a^{4}+\frac{263488}{10481}a^{3}-\frac{88520}{10481}a^{2}-\frac{11041}{223}a+\frac{40267}{10481}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 99.6310651223 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 99.6310651223 \cdot 1}{2\cdot\sqrt{4848615424}}\cr\approx \mathstrut & 0.451893014162 \end{aligned}\]
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: | data not computed |
Minimal sibling: | 8.0.5473632256.7 |
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 | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.1.0.1}{1} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.24.10 | $x^{8} + 8 x^{7} + 8 x^{6} + 2 x^{4} + 4 x^{2} + 8 x + 2$ | $8$ | $1$ | $24$ | $C_4\times C_2$ | $[2, 3, 4]$ |
\(17\) | $\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{17}$ | $x + 14$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
17.2.1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.2.1.1 | $x^{2} + 17$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |