Normalized defining polynomial
\( x^{8} - 5x^{6} - 4x^{4} + 15x^{2} + 9 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(469805625\) \(\medspace = 3^{2}\cdot 5^{4}\cdot 17^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.13\) | 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: | $3^{1/2}5^{1/2}17^{1/2}\approx 15.968719422671311$ | ||
Ramified primes: | \(3\), \(5\), \(17\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{12}a^{6}-\frac{1}{6}a^{4}-\frac{1}{3}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{24}a^{7}-\frac{1}{24}a^{6}+\frac{1}{6}a^{5}-\frac{1}{6}a^{4}-\frac{1}{6}a^{3}-\frac{1}{3}a^{2}-\frac{3}{8}a-\frac{1}{8}$
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)
| |
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: | $\frac{1}{12}a^{6}-\frac{2}{3}a^{4}+\frac{2}{3}a^{2}+\frac{9}{4}$, $\frac{1}{24}a^{7}-\frac{1}{8}a^{6}-\frac{1}{3}a^{5}+\frac{1}{2}a^{4}+\frac{1}{3}a^{3}+\frac{1}{2}a^{2}+\frac{5}{8}a-\frac{7}{8}$, $\frac{1}{24}a^{7}+\frac{1}{8}a^{6}-\frac{1}{3}a^{5}-\frac{1}{2}a^{4}+\frac{1}{3}a^{3}-\frac{1}{2}a^{2}+\frac{5}{8}a-\frac{1}{8}$, $\frac{1}{24}a^{7}+\frac{1}{8}a^{6}-\frac{1}{3}a^{5}-\frac{1}{2}a^{4}+\frac{1}{3}a^{3}-\frac{1}{2}a^{2}+\frac{13}{8}a-\frac{1}{8}$, $\frac{1}{12}a^{7}-\frac{2}{3}a^{5}+\frac{7}{6}a^{3}+\frac{1}{4}a+\frac{1}{2}$ | 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: | \( 18.2305095514 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 18.2305095514 \cdot 1}{2\cdot\sqrt{469805625}}\cr\approx \mathstrut & 0.265637524968 \end{aligned}\]
Galois group
A solvable group of order 16 |
The 10 conjugacy class representatives for $Q_8:C_2$ |
Character table for $Q_8:C_2$ |
Intermediate fields
\(\Q(\sqrt{5}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{5}, \sqrt{17})\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 16.0.17878103347812890625.1 |
Degree 8 siblings: | 8.0.169130025.1, 8.0.14630625.1 |
Minimal sibling: | 8.0.14630625.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | R | R | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | R | ${\href{/padicField/19.1.0.1}{1} }^{8}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
3.4.0.1 | $x^{4} + 2 x^{3} + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(5\) | 5.8.4.1 | $x^{8} + 80 x^{7} + 2428 x^{6} + 33688 x^{5} + 195810 x^{4} + 305952 x^{3} + 870132 x^{2} + 1037416 x + 503089$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
\(17\) | 17.8.4.1 | $x^{8} + 612 x^{7} + 140536 x^{6} + 14363966 x^{5} + 553913435 x^{4} + 345855654 x^{3} + 4032327212 x^{2} + 6379401496 x + 2294776272$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
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) | $1$ | $1$ |
1.51.2t1.a.a | $1$ | $ 3 \cdot 17 $ | \(\Q(\sqrt{-51}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.255.2t1.a.a | $1$ | $ 3 \cdot 5 \cdot 17 $ | \(\Q(\sqrt{-255}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.85.2t1.a.a | $1$ | $ 5 \cdot 17 $ | \(\Q(\sqrt{85}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.17.2t1.a.a | $1$ | $ 17 $ | \(\Q(\sqrt{17}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.15.2t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\sqrt{-15}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 2.255.8t11.b.a | $2$ | $ 3 \cdot 5 \cdot 17 $ | 8.4.469805625.1 | $Q_8:C_2$ (as 8T11) | $0$ | $0$ |
* | 2.255.8t11.b.b | $2$ | $ 3 \cdot 5 \cdot 17 $ | 8.4.469805625.1 | $Q_8:C_2$ (as 8T11) | $0$ | $0$ |