Normalized defining polynomial
\( x^{4} - 2x^{3} + 7x^{2} - 6x - 8 \)
Invariants
Degree: | $4$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 1]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-43639\) \(\medspace = -\,17^{2}\cdot 151\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(14.45\) | 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: | $17^{1/2}151^{1/2}\approx 50.665570163573605$ | ||
Ramified primes: | \(17\), \(151\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-151}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{3}-\frac{1}{2}a$
Monogenic: | No | |
Index: | $4$ | |
Inessential primes: | $2$ |
Class group and class number
$C_{3}$, which has order $3$
Unit group
Rank: | $2$ | 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: | $a^{2}-a-1$, $a^{3}+7a+5$ | 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: | \( 17.2167853413 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{1}\cdot 17.2167853413 \cdot 3}{2\cdot\sqrt{43639}}\cr\approx \mathstrut & 3.10703244547 \end{aligned}\]
Galois group
A solvable group of order 8 |
The 5 conjugacy class representatives for $D_{4}$ |
Character table for $D_{4}$ |
Intermediate fields
\(\Q(\sqrt{17}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 8.0.43421365281121.1 |
Degree 4 sibling: | 4.0.387617.1 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.1.0.1}{1} }^{4}$ | ${\href{/padicField/3.4.0.1}{4} }$ | ${\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(17\) | 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
\(151\) | 151.2.1.1 | $x^{2} + 453$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
151.2.0.1 | $x^{2} + 149 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.17.2t1.a.a | $1$ | $ 17 $ | \(\Q(\sqrt{17}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.151.2t1.a.a | $1$ | $ 151 $ | \(\Q(\sqrt{-151}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.2567.2t1.a.a | $1$ | $ 17 \cdot 151 $ | \(\Q(\sqrt{-2567}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 2.2567.4t3.b.a | $2$ | $ 17 \cdot 151 $ | 4.2.43639.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |