Normalized defining polynomial
\( x^{8} - 2x^{7} - 10x^{6} + 7x^{5} + 5x^{4} - 16x^{3} + 64x^{2} - 76x + 92 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-335111316544\) \(\medspace = -\,2^{6}\cdot 269^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(27.58\) | 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: | $2^{2}269^{1/2}\approx 65.60487786742691$ | ||
Ramified primes: | \(2\), \(269\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{172396}a^{7}+\frac{1864}{43099}a^{6}+\frac{326}{6157}a^{5}-\frac{2827}{24628}a^{4}-\frac{15381}{172396}a^{3}+\frac{4506}{43099}a^{2}+\frac{20187}{86198}a-\frac{16649}{43099}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $4$ | 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: | $\frac{559}{172396}a^{7}-\frac{12699}{172396}a^{6}+\frac{1205}{12314}a^{5}+\frac{20527}{24628}a^{4}-\frac{26869}{43099}a^{3}-\frac{311443}{172396}a^{2}+\frac{104045}{43099}a-\frac{210309}{86198}$, $\frac{5465}{86198}a^{7}-\frac{6129}{172396}a^{6}-\frac{9603}{12314}a^{5}-\frac{7799}{12314}a^{4}+\frac{101067}{172396}a^{3}+\frac{169187}{172396}a^{2}+\frac{364921}{86198}a-\frac{62283}{86198}$, $\frac{2999}{86198}a^{7}-\frac{7837}{86198}a^{6}-\frac{2578}{6157}a^{5}+\frac{6173}{12314}a^{4}+\frac{58804}{43099}a^{3}+\frac{50929}{86198}a^{2}+\frac{116015}{43099}a-\frac{129616}{43099}$, $\frac{4917}{172396}a^{7}-\frac{16097}{172396}a^{6}-\frac{1913}{12314}a^{5}+\frac{14461}{24628}a^{4}-\frac{40507}{43099}a^{3}-\frac{289229}{172396}a^{2}+\frac{173637}{43099}a-\frac{424155}{86198}$ | 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: | \( 640.591931051 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 640.591931051 \cdot 1}{2\cdot\sqrt{335111316544}}\cr\approx \mathstrut & 0.548979993459 \end{aligned}\]
Galois group
$S_4\wr C_2$ (as 8T47):
A solvable group of order 1152 |
The 20 conjugacy class representatives for $S_4\wr C_2$ |
Character table for $S_4\wr C_2$ |
Intermediate fields
\(\Q(\sqrt{269}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 12 siblings: | data not computed |
Degree 16 siblings: | data not computed |
Degree 18 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 siblings: | data not computed |
Degree 36 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 | ${\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.8.0.1}{8} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.8.0.1}{8} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.6.6 | $x^{4} - 4 x^{3} + 28 x^{2} - 24 x + 36$ | $2$ | $2$ | $6$ | $D_{4}$ | $[2, 3]^{2}$ |
2.4.0.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(269\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $6$ | $2$ | $3$ | $3$ |
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) | $1$ | $-1$ | |
1.1076.2t1.a.a | $1$ | $ 2^{2} \cdot 269 $ | \(\Q(\sqrt{-269}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.269.2t1.a.a | $1$ | $ 269 $ | \(\Q(\sqrt{269}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.17216.4t3.b.a | $2$ | $ 2^{6} \cdot 269 $ | 4.0.68864.2 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
4.68864.6t13.b.a | $4$ | $ 2^{8} \cdot 269 $ | 6.0.275456.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.74097664.12t34.b.a | $4$ | $ 2^{10} \cdot 269^{2}$ | 6.0.275456.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $-2$ | |
4.4983067904.12t34.b.a | $4$ | $ 2^{8} \cdot 269^{3}$ | 6.0.275456.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.4631104.6t13.b.a | $4$ | $ 2^{6} \cdot 269^{2}$ | 6.0.275456.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $2$ | |
6.127...424.12t201.a.a | $6$ | $ 2^{16} \cdot 269^{3}$ | 8.2.335111316544.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
6.318916345856.12t202.a.a | $6$ | $ 2^{14} \cdot 269^{3}$ | 8.2.335111316544.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
* | 6.1245766976.8t47.a.a | $6$ | $ 2^{6} \cdot 269^{3}$ | 8.2.335111316544.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |
6.79729086464.12t200.a.a | $6$ | $ 2^{12} \cdot 269^{3}$ | 8.2.335111316544.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
9.318916345856.16t1294.a.a | $9$ | $ 2^{14} \cdot 269^{3}$ | 8.2.335111316544.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
9.204...784.18t272.a.a | $9$ | $ 2^{20} \cdot 269^{3}$ | 8.2.335111316544.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.397...456.18t273.a.a | $9$ | $ 2^{20} \cdot 269^{6}$ | 8.2.335111316544.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.620...304.18t274.a.a | $9$ | $ 2^{14} \cdot 269^{6}$ | 8.2.335111316544.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
12.254...184.36t1763.a.a | $12$ | $ 2^{26} \cdot 269^{6}$ | 8.2.335111316544.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
12.158...824.24t2821.a.a | $12$ | $ 2^{22} \cdot 269^{6}$ | 8.2.335111316544.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
18.202...376.36t1758.a.a | $18$ | $ 2^{38} \cdot 269^{9}$ | 8.2.335111316544.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |