Normalized defining polynomial
\( x^{8} - x^{7} - 47x^{6} + 40x^{5} + 581x^{4} - 220x^{3} - 2038x^{2} - 932x - 109 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[8, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(9294114390625\) \(\medspace = 5^{6}\cdot 29^{6}\) | 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: | \(41.79\) | 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: | $5^{3/4}29^{3/4}\approx 41.78553833475025$ | ||
Ramified primes: | \(5\), \(29\) | 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: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5}a^{6}+\frac{1}{5}a^{5}-\frac{1}{5}a^{4}+\frac{2}{5}a^{3}+\frac{1}{5}a^{2}+\frac{1}{5}$, $\frac{1}{18723245}a^{7}+\frac{262688}{3744649}a^{6}+\frac{3599183}{18723245}a^{5}-\frac{5271837}{18723245}a^{4}+\frac{2328609}{18723245}a^{3}-\frac{7707136}{18723245}a^{2}+\frac{3778196}{18723245}a-\frac{8769786}{18723245}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
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: | $\frac{15424}{18723245}a^{7}-\frac{10506}{3744649}a^{6}-\frac{622833}{18723245}a^{5}+\frac{2239147}{18723245}a^{4}+\frac{5281306}{18723245}a^{3}-\frac{19706404}{18723245}a^{2}-\frac{10566581}{18723245}a+\frac{10265861}{18723245}$, $\frac{117117}{18723245}a^{7}-\frac{283791}{18723245}a^{6}-\frac{1175574}{3744649}a^{5}+\frac{10971787}{18723245}a^{4}+\frac{79295861}{18723245}a^{3}-\frac{78877038}{18723245}a^{2}-\frac{313763318}{18723245}a-\frac{61124328}{18723245}$, $\frac{289852}{18723245}a^{7}-\frac{274354}{18723245}a^{6}-\frac{13841888}{18723245}a^{5}+\frac{2288142}{3744649}a^{4}+\frac{35492003}{3744649}a^{3}-\frac{80890816}{18723245}a^{2}-\frac{641523388}{18723245}a-\frac{169631346}{18723245}$, $\frac{320954}{18723245}a^{7}-\frac{7883}{3744649}a^{6}-\frac{14589428}{18723245}a^{5}+\frac{2478152}{18723245}a^{4}+\frac{169111526}{18723245}a^{3}-\frac{14614569}{18723245}a^{2}-\frac{545540791}{18723245}a-\frac{107367974}{18723245}$, $\frac{89431}{18723245}a^{7}+\frac{102808}{18723245}a^{6}-\frac{4003869}{18723245}a^{5}-\frac{622340}{3744649}a^{4}+\frac{8720486}{3744649}a^{3}+\frac{9427867}{18723245}a^{2}-\frac{85449019}{18723245}a-\frac{22679153}{18723245}$, $\frac{1095}{24799}a^{7}-\frac{1205}{24799}a^{6}-\frac{50891}{24799}a^{5}+\frac{49705}{24799}a^{4}+\frac{613650}{24799}a^{3}-\frac{338215}{24799}a^{2}-\frac{2076470}{24799}a-\frac{420483}{24799}$, $\frac{1341662}{18723245}a^{7}-\frac{367126}{3744649}a^{6}-\frac{62502294}{18723245}a^{5}+\frac{75613801}{18723245}a^{4}+\frac{757030768}{18723245}a^{3}-\frac{534341762}{18723245}a^{2}-\frac{2596987623}{18723245}a-\frac{513805802}{18723245}$ | 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: | \( 3903.0092275 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 3903.0092275 \cdot 2}{2\cdot\sqrt{9294114390625}}\cr\approx \mathstrut & 0.32774459379 \end{aligned}\]
Galois group
A solvable group of order 8 |
The 5 conjugacy class representatives for $Q_8$ |
Character table for $Q_8$ |
Intermediate fields
\(\Q(\sqrt{145}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{29})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.1.0.1}{1} }^{8}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
5.4.3.2 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(29\) | 29.4.3.2 | $x^{4} + 29$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
29.4.3.2 | $x^{4} + 29$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.145.2t1.a.a | $1$ | $ 5 \cdot 29 $ | \(\Q(\sqrt{145}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.29.2t1.a.a | $1$ | $ 29 $ | \(\Q(\sqrt{29}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
*2 | 2.21025.8t5.a.a | $2$ | $ 5^{2} \cdot 29^{2}$ | 8.8.9294114390625.1 | $Q_8$ (as 8T5) | $-1$ | $2$ |