Normalized defining polynomial
\( x^{8} - 2x^{7} + 5x^{6} - 8x^{5} + 20x^{4} - 36x^{3} + 65x^{2} - 66x + 41 \)
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: | \(534534400\) \(\medspace = 2^{8}\cdot 5^{2}\cdot 17^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.33\) | 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: | $2\cdot 5^{1/2}17^{3/4}\approx 37.44136633070439$ | ||
Ramified primes: | \(2\), \(5\), \(17\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6}a^{6}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a+\frac{1}{6}$, $\frac{1}{318}a^{7}-\frac{17}{318}a^{6}+\frac{8}{53}a^{5}+\frac{20}{53}a^{4}+\frac{11}{159}a^{3}-\frac{77}{159}a^{2}+\frac{149}{318}a+\frac{137}{318}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $3$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{19}{106} a^{7} - \frac{5}{106} a^{6} + \frac{32}{53} a^{5} - \frac{26}{53} a^{4} + \frac{103}{53} a^{3} - \frac{138}{53} a^{2} + \frac{499}{106} a - \frac{153}{106} \) (order $4$) | 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: | $\frac{13}{159}a^{7}-\frac{71}{318}a^{6}+\frac{41}{159}a^{5}-\frac{136}{159}a^{4}+\frac{60}{53}a^{3}-\frac{571}{159}a^{2}+\frac{718}{159}a-\frac{491}{106}$, $\frac{49}{318}a^{7}-\frac{19}{159}a^{6}+\frac{21}{53}a^{5}-\frac{27}{53}a^{4}+\frac{221}{159}a^{3}-\frac{434}{159}a^{2}+\frac{1259}{318}a-\frac{380}{159}$, $\frac{3}{53}a^{7}-\frac{41}{318}a^{6}+\frac{61}{159}a^{5}-\frac{86}{159}a^{4}+\frac{145}{159}a^{3}-\frac{144}{53}a^{2}+\frac{811}{159}a-\frac{1403}{318}$ | 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: | \( 16.4873901267 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 16.4873901267 \cdot 2}{4\cdot\sqrt{534534400}}\cr\approx \mathstrut & 0.555716846285 \end{aligned}\]
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
\(\Q(\sqrt{-1}) \), 4.0.272.1 |
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: | 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 | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | 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 | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.1.0.1}{1} }^{8}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ |
2.4.4.1 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
\(5\) | 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(17\) | 17.2.1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
17.4.3.1 | $x^{4} + 17$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.68.2t1.a.a | $1$ | $ 2^{2} \cdot 17 $ | \(\Q(\sqrt{-17}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.17.2t1.a.a | $1$ | $ 17 $ | \(\Q(\sqrt{17}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.340.4t1.c.a | $1$ | $ 2^{2} \cdot 5 \cdot 17 $ | 4.0.1965200.1 | $C_4$ (as 4T1) | $0$ | $-1$ | |
1.85.4t1.a.a | $1$ | $ 5 \cdot 17 $ | 4.4.122825.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
1.85.4t1.a.b | $1$ | $ 5 \cdot 17 $ | 4.4.122825.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
1.340.4t1.c.b | $1$ | $ 2^{2} \cdot 5 \cdot 17 $ | 4.0.1965200.1 | $C_4$ (as 4T1) | $0$ | $-1$ | |
2.28900.4t3.a.a | $2$ | $ 2^{2} \cdot 5^{2} \cdot 17^{2}$ | 4.0.1965200.2 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
* | 2.68.4t3.a.a | $2$ | $ 2^{2} \cdot 17 $ | 4.0.272.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
* | 4.1965200.8t19.d.a | $4$ | $ 2^{4} \cdot 5^{2} \cdot 17^{3}$ | 8.0.534534400.3 | $C_2^3 : C_4 $ (as 8T19) | $1$ | $0$ |