Normalized defining polynomial
\( x^{8} - x^{7} - 219x^{6} + 186x^{5} + 8843x^{4} - 598x^{3} - 96824x^{2} - 92144x - 14521 \)
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)
| |
Discriminant: | \(106983137776707361\) \(\medspace = 13^{6}\cdot 53^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(134.48\) | 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))
| |
Galois root discriminant: | $13^{3/4}53^{3/4}\approx 134.48207963630105$ | ||
Ramified primes: | \(13\), \(53\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{29}a^{5}+\frac{1}{29}a^{4}-\frac{1}{29}a^{3}+\frac{3}{29}a^{2}+\frac{12}{29}$, $\frac{1}{377}a^{6}+\frac{6}{377}a^{5}+\frac{178}{377}a^{4}+\frac{11}{29}a^{3}-\frac{10}{29}a^{2}-\frac{8}{29}a-\frac{11}{29}$, $\frac{1}{190007671633}a^{7}+\frac{74486392}{190007671633}a^{6}+\frac{167977624}{190007671633}a^{5}+\frac{411346227}{14615974741}a^{4}-\frac{2422999456}{14615974741}a^{3}-\frac{1820736829}{14615974741}a^{2}-\frac{5906115406}{14615974741}a-\frac{871605389}{14615974741}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{4}$, which has order $4$
Unit group
Rank: | $7$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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)
| |
Fundamental units: | $\frac{4966550}{190007671633}a^{7}-\frac{10489690}{190007671633}a^{6}-\frac{993755629}{190007671633}a^{5}+\frac{2114216043}{190007671633}a^{4}+\frac{1959000654}{14615974741}a^{3}-\frac{3933593758}{14615974741}a^{2}-\frac{5578323091}{14615974741}a+\frac{3486196887}{14615974741}$, $\frac{2761}{17054813}a^{7}-\frac{4491}{17054813}a^{6}-\frac{600567}{17054813}a^{5}+\frac{887509}{17054813}a^{4}+\frac{23483902}{17054813}a^{3}-\frac{15975297}{17054813}a^{2}-\frac{235063816}{17054813}a-\frac{48350864}{17054813}$, $\frac{81386802}{190007671633}a^{7}-\frac{142027222}{190007671633}a^{6}-\frac{598512290}{6551988677}a^{5}+\frac{28232339710}{190007671633}a^{4}+\frac{48087410644}{14615974741}a^{3}-\frac{43116034090}{14615974741}a^{2}-\frac{425212672988}{14615974741}a-\frac{83544568089}{14615974741}$, $\frac{36240216}{190007671633}a^{7}-\frac{9372800}{6551988677}a^{6}-\frac{545746266}{14615974741}a^{5}+\frac{52899820630}{190007671633}a^{4}+\frac{12347341131}{14615974741}a^{3}-\frac{81521864467}{14615974741}a^{2}-\frac{87779442919}{14615974741}a-\frac{14024334896}{14615974741}$, $\frac{3053338}{190007671633}a^{7}+\frac{4219601}{190007671633}a^{6}-\frac{56855427}{14615974741}a^{5}-\frac{292061680}{190007671633}a^{4}+\frac{2685545031}{14615974741}a^{3}+\frac{148513752}{14615974741}a^{2}-\frac{28790335502}{14615974741}a-\frac{467256007}{503999129}$, $\frac{7052896}{14615974741}a^{7}+\frac{61043809}{190007671633}a^{6}-\frac{19402297178}{190007671633}a^{5}-\frac{13263923341}{190007671633}a^{4}+\frac{51931576100}{14615974741}a^{3}+\frac{1707501851}{503999129}a^{2}-\frac{382279023386}{14615974741}a-\frac{319057070170}{14615974741}$, $\frac{137997849}{190007671633}a^{7}-\frac{568356650}{190007671633}a^{6}-\frac{2178131684}{14615974741}a^{5}+\frac{113372900211}{190007671633}a^{4}+\frac{64639201525}{14615974741}a^{3}-\frac{199094777201}{14615974741}a^{2}-\frac{355371840324}{14615974741}a-\frac{71091313629}{14615974741}$ | 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: | \( 140299.139687 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 140299.139687 \cdot 4}{2\cdot\sqrt{106983137776707361}}\cr\approx \mathstrut & 0.219617681908 \end{aligned}\]
Galois group
A solvable group of order 8 |
The 5 conjugacy class representatives for $Q_8$ |
Character table for $Q_8$ |
Intermediate fields
\(\Q(\sqrt{689}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{13}, \sqrt{53})\) |
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 | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.1.0.1}{1} }^{8}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(13\) | 13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
13.4.3.2 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(53\) | 53.4.3.2 | $x^{4} + 53$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
53.4.3.2 | $x^{4} + 53$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.689.2t1.a.a | $1$ | $ 13 \cdot 53 $ | \(\Q(\sqrt{689}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.53.2t1.a.a | $1$ | $ 53 $ | \(\Q(\sqrt{53}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.13.2t1.a.a | $1$ | $ 13 $ | \(\Q(\sqrt{13}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
*2 | 2.474721.8t5.a.a | $2$ | $ 13^{2} \cdot 53^{2}$ | 8.8.106983137776707361.1 | $Q_8$ (as 8T5) | $-1$ | $2$ |