Normalized defining polynomial
\( x^{8} - x^{7} + 7x^{6} - 8x^{5} + 11x^{4} + 5x^{3} + 11x^{2} - 2x + 2 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(1368408064\) \(\medspace = 2^{14}\cdot 17^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(13.87\) | 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: | $2^{11/4}17^{1/2}\approx 27.736837922354518$ | ||
Ramified primes: | \(2\), \(17\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{17}a^{6}+\frac{3}{17}a^{5}-\frac{2}{17}a^{4}+\frac{6}{17}a^{3}-\frac{8}{17}a^{2}-\frac{8}{17}$, $\frac{1}{119}a^{7}-\frac{3}{119}a^{6}+\frac{48}{119}a^{5}+\frac{1}{119}a^{4}+\frac{58}{119}a^{3}-\frac{20}{119}a^{2}+\frac{9}{119}a-\frac{20}{119}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $3$ | 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: | $\frac{11}{119}a^{7}-\frac{19}{119}a^{6}+\frac{94}{119}a^{5}-\frac{8}{7}a^{4}+\frac{246}{119}a^{3}+\frac{25}{119}a^{2}+\frac{99}{119}a+\frac{25}{119}$, $\frac{13}{119}a^{7}-\frac{18}{119}a^{6}+\frac{92}{119}a^{5}-\frac{148}{119}a^{4}+\frac{166}{119}a^{3}+\frac{48}{119}a^{2}+\frac{117}{119}a-\frac{71}{119}$, $\frac{60}{119}a^{7}-\frac{110}{119}a^{6}+\frac{472}{119}a^{5}-\frac{794}{119}a^{4}+\frac{1044}{119}a^{3}-\frac{94}{119}a^{2}+\frac{64}{119}a-\frac{927}{119}$ | 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: | \( 56.0168760267 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 56.0168760267 \cdot 1}{2\cdot\sqrt{1368408064}}\cr\approx \mathstrut & 1.18005038415 \end{aligned}\]
Galois group
$C_2^3:C_4$ (as 8T19):
A solvable group of order 32 |
The 11 conjugacy class representatives for $C_2^3 : C_4 $ |
Character table for $C_2^3 : C_4 $ |
Intermediate fields
\(\Q(\sqrt{17}) \), 4.0.2312.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | data not computed |
Degree 8 siblings: | data not computed |
Degree 16 siblings: | 16.0.119842600295694598144.9, some data not computed |
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 | R | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.2.0.1}{2} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.3.2 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ |
2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
2.4.11.3 | $x^{4} + 4 x^{2} + 18$ | $4$ | $1$ | $11$ | $C_4$ | $[3, 4]$ | |
\(17\) | 17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.17.2t1.a.a | $1$ | $ 17 $ | \(\Q(\sqrt{17}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
1.136.2t1.a.a | $1$ | $ 2^{3} \cdot 17 $ | \(\Q(\sqrt{34}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.8.2t1.a.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{2}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.272.4t1.b.a | $1$ | $ 2^{4} \cdot 17 $ | 4.0.591872.5 | $C_4$ (as 4T1) | $0$ | $-1$ | |
1.16.4t1.b.a | $1$ | $ 2^{4}$ | 4.0.2048.2 | $C_4$ (as 4T1) | $0$ | $-1$ | |
1.16.4t1.b.b | $1$ | $ 2^{4}$ | 4.0.2048.2 | $C_4$ (as 4T1) | $0$ | $-1$ | |
1.272.4t1.b.b | $1$ | $ 2^{4} \cdot 17 $ | 4.0.591872.5 | $C_4$ (as 4T1) | $0$ | $-1$ | |
* | 2.136.4t3.a.a | $2$ | $ 2^{3} \cdot 17 $ | 4.0.1088.2 | $D_{4}$ (as 4T3) | $1$ | $-2$ |
2.4352.4t3.a.a | $2$ | $ 2^{8} \cdot 17 $ | 4.4.591872.1 | $D_{4}$ (as 4T3) | $1$ | $2$ | |
* | 4.591872.8t19.a.a | $4$ | $ 2^{11} \cdot 17^{2}$ | 8.0.1368408064.2 | $C_2^3 : C_4 $ (as 8T19) | $1$ | $0$ |