Normalized defining polynomial
\( x^{8} - 2x^{6} - 7x^{5} + 10x^{4} + 7x^{3} - 12x^{2} - x + 5 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-373837707\) \(\medspace = -\,3^{3}\cdot 61^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.79\) | 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))
| |
Galois root discriminant: | $3^{1/2}61^{1/2}\approx 13.527749258468683$ | ||
Ramified primes: | \(3\), \(61\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{25}a^{7}+\frac{12}{25}a^{6}-\frac{8}{25}a^{5}-\frac{3}{25}a^{4}-\frac{1}{25}a^{3}-\frac{1}{5}a^{2}+\frac{3}{25}a+\frac{2}{5}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $4$ | 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: | $\frac{2}{25}a^{7}-\frac{1}{25}a^{6}-\frac{16}{25}a^{5}-\frac{31}{25}a^{4}+\frac{23}{25}a^{3}+\frac{18}{5}a^{2}-\frac{19}{25}a-\frac{11}{5}$, $\frac{9}{25}a^{7}+\frac{8}{25}a^{6}+\frac{3}{25}a^{5}-\frac{52}{25}a^{4}+\frac{41}{25}a^{3}+\frac{1}{5}a^{2}-\frac{23}{25}a-\frac{2}{5}$, $a-1$, $\frac{12}{25}a^{7}+\frac{19}{25}a^{6}+\frac{4}{25}a^{5}-\frac{86}{25}a^{4}-\frac{12}{25}a^{3}+\frac{18}{5}a^{2}-\frac{14}{25}a-\frac{11}{5}$ | 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: | \( 19.2565937702 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 19.2565937702 \cdot 1}{2\cdot\sqrt{373837707}}\cr\approx \mathstrut & 0.494091653323 \end{aligned}\]
Galois group
A solvable group of order 16 |
The 7 conjugacy class representatives for $D_{8}$ |
Character table for $D_{8}$ |
Intermediate fields
\(\Q(\sqrt{61}) \), 4.2.11163.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 16.0.1257791680575160641.1 |
Degree 8 sibling: | 8.0.18385461.1 |
Minimal sibling: | 8.0.18385461.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.8.0.1}{8} }$ | R | ${\href{/padicField/5.2.0.1}{2} }^{3}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.2.0.1}{2} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.4.2.1 | $x^{4} + 4 x^{3} + 14 x^{2} + 20 x + 13$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(61\) | 61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
61.4.2.1 | $x^{4} + 4878 x^{3} + 6091587 x^{2} + 348450174 x + 20534983$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.61.2t1.a.a | $1$ | $ 61 $ | \(\Q(\sqrt{61}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.183.2t1.a.a | $1$ | $ 3 \cdot 61 $ | \(\Q(\sqrt{-183}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 2.183.4t3.a.a | $2$ | $ 3 \cdot 61 $ | 4.0.549.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
* | 2.183.8t6.a.a | $2$ | $ 3 \cdot 61 $ | 8.2.373837707.1 | $D_{8}$ (as 8T6) | $1$ | $0$ |
* | 2.183.8t6.a.b | $2$ | $ 3 \cdot 61 $ | 8.2.373837707.1 | $D_{8}$ (as 8T6) | $1$ | $0$ |