Normalized defining polynomial
\( x^{8} - x^{7} - 7x^{6} + 6x^{5} - 19x^{4} + 24x^{3} + 58x^{2} + 106x - 67 \)
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: | \(148132260953\) \(\medspace = 17^{7}\cdot 19^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(24.91\) | 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: | $17^{7/8}19^{1/2}\approx 52.00194728311084$ | ||
Ramified primes: | \(17\), \(19\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{17}) \) | ||
$\card{ \Aut(K/\Q) }$: | $4$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{512291}a^{7}-\frac{200982}{512291}a^{6}-\frac{69724}{512291}a^{5}-\frac{8764}{512291}a^{4}+\frac{141007}{512291}a^{3}+\frac{210277}{512291}a^{2}-\frac{235634}{512291}a+\frac{240147}{512291}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
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: | $\frac{1639}{512291}a^{7}-\frac{6385}{512291}a^{6}-\frac{36743}{512291}a^{5}-\frac{20048}{512291}a^{4}+\frac{67232}{512291}a^{3}+\frac{384451}{512291}a^{2}+\frac{575579}{512291}a+\frac{673736}{512291}$, $\frac{5776}{512291}a^{7}-\frac{20626}{512291}a^{6}-\frac{65098}{512291}a^{5}+\frac{95945}{512291}a^{4}-\frac{86258}{512291}a^{3}+\frac{430282}{512291}a^{2}+\frac{1159785}{512291}a+\frac{829626}{512291}$, $\frac{1932}{512291}a^{7}+\frac{19354}{512291}a^{6}+\frac{25765}{512291}a^{5}-\frac{26445}{512291}a^{4}-\frac{113288}{512291}a^{3}-\frac{503890}{512291}a^{2}-\frac{842771}{512291}a-\frac{683933}{512291}$, $\frac{7142}{512291}a^{7}+\frac{25938}{512291}a^{6}-\frac{21956}{512291}a^{5}-\frac{92986}{512291}a^{4}-\frac{92112}{512291}a^{3}-\frac{751169}{512291}a^{2}-\frac{534384}{512291}a+\frac{491897}{512291}$, $\frac{719}{39407}a^{7}-\frac{589}{39407}a^{6}-\frac{5852}{39407}a^{5}+\frac{3804}{39407}a^{4}-\frac{10178}{39407}a^{3}+\frac{23911}{39407}a^{2}+\frac{29254}{39407}a+\frac{63033}{39407}$ | 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: | \( 177.70397323 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{2}\cdot 177.70397323 \cdot 2}{2\cdot\sqrt{148132260953}}\cr\approx \mathstrut & 0.29164331884 \end{aligned}\]
Galois group
$\OD_{16}$ (as 8T7):
A solvable group of order 16 |
The 10 conjugacy class representatives for $C_8:C_2$ |
Character table for $C_8:C_2$ |
Intermediate fields
\(\Q(\sqrt{17}) \), 4.4.4913.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 16.0.2859655432078149808865465089.5 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | ${\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.8.0.1}{8} }$ | ${\href{/padicField/11.8.0.1}{8} }$ | ${\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | R | R | ${\href{/padicField/23.8.0.1}{8} }$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.8.0.1}{8} }$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.8.0.1}{8} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.1.0.1}{1} }^{8}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(17\) | 17.8.7.1 | $x^{8} + 68$ | $8$ | $1$ | $7$ | $C_8$ | $[\ ]_{8}$ |
\(19\) | 19.4.0.1 | $x^{4} + 2 x^{2} + 11 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
19.4.2.2 | $x^{4} - 2888 x^{3} - 767106 x^{2} - 76532 x + 722$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.19.2t1.a.a | $1$ | $ 19 $ | \(\Q(\sqrt{-19}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.323.2t1.a.a | $1$ | $ 17 \cdot 19 $ | \(\Q(\sqrt{-323}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.17.2t1.a.a | $1$ | $ 17 $ | \(\Q(\sqrt{17}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
* | 1.17.4t1.a.a | $1$ | $ 17 $ | 4.4.4913.1 | $C_4$ (as 4T1) | $0$ | $1$ |
1.323.4t1.a.a | $1$ | $ 17 \cdot 19 $ | 4.0.1773593.2 | $C_4$ (as 4T1) | $0$ | $-1$ | |
1.323.4t1.a.b | $1$ | $ 17 \cdot 19 $ | 4.0.1773593.2 | $C_4$ (as 4T1) | $0$ | $-1$ | |
* | 1.17.4t1.a.b | $1$ | $ 17 $ | 4.4.4913.1 | $C_4$ (as 4T1) | $0$ | $1$ |
* | 2.5491.8t7.a.a | $2$ | $ 17^{2} \cdot 19 $ | 8.4.148132260953.1 | $C_8:C_2$ (as 8T7) | $0$ | $0$ |
* | 2.5491.8t7.a.b | $2$ | $ 17^{2} \cdot 19 $ | 8.4.148132260953.1 | $C_8:C_2$ (as 8T7) | $0$ | $0$ |