Normalized defining polynomial
\( x^{8} - 44x^{6} + 308x^{4} - 484x^{2} + 121 \)
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: | \(29721861554176\) \(\medspace = 2^{24}\cdot 11^{6}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(48.32\) | 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: | $2^{3}11^{3/4}\approx 48.32084283629789$ | ||
Ramified primes: | \(2\), \(11\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{33}a^{4}+\frac{1}{3}a^{2}+\frac{1}{3}$, $\frac{1}{33}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{231}a^{6}-\frac{1}{3}a^{2}-\frac{2}{21}$, $\frac{1}{231}a^{7}-\frac{1}{3}a^{3}-\frac{2}{21}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $7$ |
Class group and class number
Trivial group, which has order $1$
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: | $\frac{1}{77}a^{6}-\frac{6}{11}a^{4}+3a^{2}-\frac{9}{7}$, $\frac{1}{231}a^{6}-\frac{5}{33}a^{4}+\frac{5}{21}$, $\frac{17}{231}a^{7}+\frac{1}{33}a^{6}-\frac{107}{33}a^{5}-\frac{46}{33}a^{4}+\frac{68}{3}a^{3}+\frac{34}{3}a^{2}-\frac{219}{7}a-17$, $\frac{4}{231}a^{7}+\frac{10}{231}a^{6}-\frac{25}{33}a^{5}-\frac{64}{33}a^{4}+\frac{16}{3}a^{3}+\frac{43}{3}a^{2}-\frac{96}{7}a-\frac{79}{7}$, $\frac{1}{77}a^{7}-\frac{1}{231}a^{6}-\frac{20}{33}a^{5}+\frac{3}{11}a^{4}+\frac{16}{3}a^{3}-\frac{14}{3}a^{2}-\frac{167}{21}a+\frac{149}{21}$, $\frac{4}{77}a^{7}+\frac{4}{231}a^{6}-\frac{74}{33}a^{5}-\frac{9}{11}a^{4}+\frac{43}{3}a^{3}+\frac{20}{3}a^{2}-\frac{332}{21}a-\frac{281}{21}$, $\frac{61}{231}a^{7}-\frac{26}{77}a^{6}-\frac{123}{11}a^{5}+\frac{43}{3}a^{4}+\frac{188}{3}a^{3}-\frac{244}{3}a^{2}-\frac{395}{21}a+\frac{485}{21}$ | 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: | \( 43308.6570657 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 43308.6570657 \cdot 1}{2\cdot\sqrt{29721861554176}}\cr\approx \mathstrut & 1.01682609564 \end{aligned}\]
Galois group
A solvable group of order 8 |
The 5 conjugacy class representatives for $Q_8$ |
Character table for $Q_8$ |
Intermediate fields
\(\Q(\sqrt{2}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{22}) \), \(\Q(\sqrt{2}, \sqrt{11})\) |
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 | R | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.1.0.1}{1} }^{8}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.24.4 | $x^{8} + 4 x^{6} + 10 x^{4} + 4 x^{2} + 8 x + 30$ | $8$ | $1$ | $24$ | $Q_8$ | $[2, 3, 4]$ |
\(11\) | 11.8.6.1 | $x^{8} - 110 x^{4} - 16819$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.88.2t1.a.a | $1$ | $ 2^{3} \cdot 11 $ | \(\Q(\sqrt{22}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.44.2t1.a.a | $1$ | $ 2^{2} \cdot 11 $ | \(\Q(\sqrt{11}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
*2 | 2.30976.8t5.a.a | $2$ | $ 2^{8} \cdot 11^{2}$ | 8.8.29721861554176.1 | $Q_8$ (as 8T5) | $-1$ | $2$ |