Normalized defining polynomial
\( x^{8} - 132x^{6} + 5940x^{4} - 100188x^{2} + 393129 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[8, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(21667237072994304\) \(\medspace = 2^{24}\cdot 3^{6}\cdot 11^{6}\) | 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: | \(110.15\) | 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: | $2^{3}3^{3/4}11^{3/4}\approx 110.14770224334376$ | ||
Ramified primes: | \(2\), \(3\), \(11\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{165}a^{4}-\frac{2}{5}$, $\frac{1}{165}a^{5}-\frac{2}{5}a$, $\frac{1}{825}a^{6}-\frac{1}{825}a^{4}-\frac{2}{25}a^{2}+\frac{2}{25}$, $\frac{1}{15675}a^{7}+\frac{13}{5225}a^{5}+\frac{123}{475}a^{3}-\frac{53}{475}a$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $5$ |
Class group and class number
$C_{2}\times C_{4}$, which has order $8$
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)
| |
Fundamental units: | $\frac{1}{825}a^{6}-\frac{76}{825}a^{4}+\frac{48}{25}a^{2}-\frac{198}{25}$, $\frac{1}{275}a^{6}-\frac{298}{825}a^{4}+\frac{244}{25}a^{2}-\frac{1104}{25}$, $\frac{1}{825}a^{6}-\frac{37}{275}a^{4}+\frac{123}{25}a^{2}-\frac{1478}{25}$, $\frac{46}{15675}a^{7}+\frac{2}{55}a^{6}-\frac{4381}{15675}a^{5}-\frac{527}{165}a^{4}+\frac{3283}{475}a^{3}+\frac{378}{5}a^{2}-\frac{14313}{475}a-\frac{1631}{5}$, $\frac{47}{15675}a^{7}-\frac{13}{825}a^{6}-\frac{4627}{15675}a^{5}+\frac{1223}{825}a^{4}+\frac{3881}{475}a^{3}-\frac{924}{25}a^{2}-\frac{23296}{475}a+\frac{3304}{25}$, $\frac{3}{475}a^{7}+\frac{2}{55}a^{6}-\frac{884}{1425}a^{5}-\frac{587}{165}a^{4}+\frac{7902}{475}a^{3}+\frac{473}{5}a^{2}-\frac{36027}{475}a-\frac{2061}{5}$, $\frac{109}{15675}a^{7}+\frac{37}{825}a^{6}-\frac{9904}{15675}a^{5}-\frac{3397}{825}a^{4}+\frac{7232}{475}a^{3}+\frac{2526}{25}a^{2}-\frac{29717}{475}a-\frac{11181}{25}$ | 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: | \( 373376.024336 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{8}\cdot(2\pi)^{0}\cdot 373376.024336 \cdot 8}{2\cdot\sqrt{21667237072994304}}\cr\approx \mathstrut & 2.59743456838 \end{aligned}\]
Galois group
A solvable group of order 8 |
The 5 conjugacy class representatives for $Q_8$ |
Character table for $Q_8$ |
Intermediate fields
\(\Q(\sqrt{66}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{6}, \sqrt{11})\) |
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 | R | ${\href{/padicField/5.1.0.1}{1} }^{8}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.1.0.1}{1} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.24.12 | $x^{8} + 4 x^{6} + 10 x^{4} + 4 x^{2} + 8 x + 14$ | $8$ | $1$ | $24$ | $Q_8$ | $[2, 3, 4]$ |
\(3\) | 3.8.6.1 | $x^{8} + 9$ | $4$ | $2$ | $6$ | $Q_8$ | $[\ ]_{4}^{2}$ |
\(11\) | 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 | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.44.2t1.a.a | $1$ | $ 2^{2} \cdot 11 $ | \(\Q(\sqrt{11}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.24.2t1.a.a | $1$ | $ 2^{3} \cdot 3 $ | \(\Q(\sqrt{6}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.264.2t1.a.a | $1$ | $ 2^{3} \cdot 3 \cdot 11 $ | \(\Q(\sqrt{66}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
*2 | 2.278784.8t5.g.a | $2$ | $ 2^{8} \cdot 3^{2} \cdot 11^{2}$ | 8.8.21667237072994304.2 | $Q_8$ (as 8T5) | $-1$ | $2$ |