Normalized defining polynomial
\( x^{8} - x^{7} - 6x^{6} + 20x^{5} - 32x^{4} - 51x^{3} + 228x^{2} - 240x + 129 \)
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)
| |
Discriminant: | \(10093618089\) \(\medspace = 3^{6}\cdot 61^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(17.80\) | 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: | $3^{3/4}61^{3/4}\approx 49.755181783405924$ | ||
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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{197080}a^{7}+\frac{222}{24635}a^{6}+\frac{1333}{98540}a^{5}+\frac{3791}{98540}a^{4}+\frac{35871}{98540}a^{3}-\frac{25277}{197080}a^{2}+\frac{17239}{197080}a+\frac{86063}{197080}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
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: | \( -\frac{271}{15160} a^{7} + \frac{17}{7580} a^{6} + \frac{351}{3790} a^{5} - \frac{1083}{3790} a^{4} + \frac{551}{1895} a^{3} + \frac{16697}{15160} a^{2} - \frac{40389}{15160} a + \frac{34697}{15160} \) (order $6$) | 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: | $\frac{3631}{197080}a^{7}-\frac{2857}{98540}a^{6}-\frac{3243}{24635}a^{5}+\frac{21713}{49270}a^{4}-\frac{23457}{49270}a^{3}-\frac{187857}{197080}a^{2}+\frac{908769}{197080}a-\frac{911717}{197080}$, $\frac{1329}{197080}a^{7}-\frac{582}{24635}a^{6}-\frac{2163}{98540}a^{5}+\frac{12699}{98540}a^{4}-\frac{20801}{98540}a^{3}+\frac{107547}{197080}a^{2}-\frac{147729}{197080}a+\frac{71327}{197080}$, $\frac{211}{15160}a^{7}-\frac{237}{7580}a^{6}-\frac{273}{1895}a^{5}+\frac{1053}{3790}a^{4}-\frac{2767}{3790}a^{3}-\frac{31237}{15160}a^{2}+\frac{44509}{15160}a-\frac{36497}{15160}$ | 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: | \( 131.269561521 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 131.269561521 \cdot 2}{6\cdot\sqrt{10093618089}}\cr\approx \mathstrut & 0.678795288307 \end{aligned}\]
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
\(\Q(\sqrt{-3}) \), 4.0.549.1 |
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: | 16.0.379099670317033992358041.4, 16.4.1410629873249683485564270561.4, 16.0.1410629873249683485564270561.3, 16.0.1410629873249683485564270561.4 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | R | ${\href{/padicField/5.2.0.1}{2} }^{4}$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ |
3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $[\ ]_{4}^{2}$ | |
\(61\) | $\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{61}$ | $x + 59$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
61.2.1.1 | $x^{2} + 61$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
61.4.3.2 | $x^{4} + 61$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.183.2t1.a.a | $1$ | $ 3 \cdot 61 $ | \(\Q(\sqrt{-183}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.61.2t1.a.a | $1$ | $ 61 $ | \(\Q(\sqrt{61}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
1.183.4t1.a.a | $1$ | $ 3 \cdot 61 $ | 4.4.2042829.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
1.61.4t1.a.a | $1$ | $ 61 $ | 4.0.226981.1 | $C_4$ (as 4T1) | $0$ | $-1$ | |
1.61.4t1.a.b | $1$ | $ 61 $ | 4.0.226981.1 | $C_4$ (as 4T1) | $0$ | $-1$ | |
1.183.4t1.a.b | $1$ | $ 3 \cdot 61 $ | 4.4.2042829.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
2.11163.4t3.b.a | $2$ | $ 3 \cdot 61^{2}$ | 4.2.680943.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
* | 2.183.4t3.a.a | $2$ | $ 3 \cdot 61 $ | 4.0.549.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
* | 4.18385461.8t19.b.a | $4$ | $ 3^{4} \cdot 61^{3}$ | 8.0.10093618089.2 | $C_2^3 : C_4 $ (as 8T19) | $1$ | $0$ |