Normalized defining polynomial
\( x^{8} - 5x^{6} - 11x^{4} - 48x^{3} + 36x^{2} + 240x + 212 \)
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: | \(3613452544\) \(\medspace = 2^{8}\cdot 13^{2}\cdot 17^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(15.66\) | 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: | $2\cdot 13^{1/2}17^{3/4}\approx 60.37238916132119$ | ||
Ramified primes: | \(2\), \(13\), \(17\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{12}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{5}{12}a^{2}-\frac{1}{2}a+\frac{1}{6}$, $\frac{1}{17448}a^{7}+\frac{86}{2181}a^{6}-\frac{2645}{5816}a^{5}-\frac{565}{1454}a^{4}+\frac{7051}{17448}a^{3}-\frac{2057}{4362}a^{2}+\frac{4159}{8724}a-\frac{2161}{4362}$
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: | \( -\frac{68}{727} a^{7} + \frac{215}{1454} a^{6} + \frac{146}{727} a^{5} - \frac{483}{1454} a^{4} + \frac{1079}{727} a^{3} + \frac{3063}{1454} a^{2} - \frac{4380}{727} a - \frac{8348}{727} \) (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: | $\frac{2239}{17448}a^{7}-\frac{929}{4362}a^{6}-\frac{1467}{5816}a^{5}+\frac{336}{727}a^{4}-\frac{38147}{17448}a^{3}-\frac{5128}{2181}a^{2}+\frac{73285}{8724}a+\frac{64409}{4362}$, $\frac{511}{17448}a^{7}-\frac{877}{8724}a^{6}+\frac{625}{5816}a^{5}+\frac{535}{2908}a^{4}-\frac{17399}{17448}a^{3}+\frac{11135}{8724}a^{2}+\frac{9679}{8724}a-\frac{5795}{2181}$, $\frac{251}{5816}a^{7}+\frac{947}{8724}a^{6}+\frac{295}{5816}a^{5}+\frac{427}{2908}a^{4}-\frac{1171}{5816}a^{3}-\frac{27721}{8724}a^{2}-\frac{18965}{2908}a-\frac{9191}{2181}$ | 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: | \( 44.1429509142 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 44.1429509142 \cdot 2}{4\cdot\sqrt{3613452544}}\cr\approx \mathstrut & 0.572255087052 \end{aligned}\]
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 | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}$ | R | R | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.2.0.1}{2} }^{4}$ |
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 | $x^{4} + 6 x^{3} + 17 x^{2} + 24 x + 13$ | $2$ | $2$ | $4$ | $C_2^2$ | $[2]^{2}$ | |
\(13\) | 13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
13.2.0.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
13.4.2.1 | $x^{4} + 284 x^{3} + 21754 x^{2} + 225780 x + 59193$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(17\) | 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
17.2.1.2 | $x^{2} + 51$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
17.4.3.2 | $x^{4} + 34$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |
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) | $1$ | $1$ | |
1.221.4t1.a.a | $1$ | $ 13 \cdot 17 $ | 4.4.830297.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
1.884.4t1.a.a | $1$ | $ 2^{2} \cdot 13 \cdot 17 $ | 4.0.13284752.4 | $C_4$ (as 4T1) | $0$ | $-1$ | |
1.884.4t1.a.b | $1$ | $ 2^{2} \cdot 13 \cdot 17 $ | 4.0.13284752.4 | $C_4$ (as 4T1) | $0$ | $-1$ | |
1.221.4t1.a.b | $1$ | $ 13 \cdot 17 $ | 4.4.830297.1 | $C_4$ (as 4T1) | $0$ | $1$ | |
* | 2.68.4t3.b.a | $2$ | $ 2^{2} \cdot 17 $ | 4.2.1156.1 | $D_{4}$ (as 4T3) | $1$ | $0$ |
2.195364.4t3.a.a | $2$ | $ 2^{2} \cdot 13^{2} \cdot 17^{2}$ | 4.0.13284752.3 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
* | 4.13284752.8t19.j.a | $4$ | $ 2^{4} \cdot 13^{2} \cdot 17^{3}$ | 8.0.3613452544.2 | $C_2^3 : C_4 $ (as 8T19) | $1$ | $0$ |