Normalized defining polynomial
\( x^{8} - 2x^{7} + x^{6} + 6x^{5} - 9x^{4} - 12x^{3} + 9x^{2} + 12x + 3 \)
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: | \(203233536\) \(\medspace = 2^{8}\cdot 3^{8}\cdot 11^{2}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(10.93\) | 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: | $2^{7/4}3^{7/6}11^{1/2}\approx 40.19206303367818$ | ||
Ramified primes: | \(2\), \(3\), \(11\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{21}a^{7}+\frac{2}{21}a^{6}+\frac{3}{7}a^{5}-\frac{3}{7}a^{3}-\frac{2}{7}a^{2}+\frac{2}{7}a-\frac{2}{7}$
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: | \( \frac{32}{21} a^{7} - \frac{83}{21} a^{6} + \frac{26}{7} a^{5} + 7 a^{4} - \frac{124}{7} a^{3} - \frac{64}{7} a^{2} + \frac{134}{7} a + \frac{69}{7} \) (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: | $\frac{13}{21}a^{7}-\frac{37}{21}a^{6}+\frac{11}{7}a^{5}+3a^{4}-\frac{60}{7}a^{3}-\frac{26}{7}a^{2}+\frac{68}{7}a+\frac{30}{7}$, $\frac{5}{21}a^{7}-\frac{11}{21}a^{6}+\frac{1}{7}a^{5}+2a^{4}-\frac{22}{7}a^{3}-\frac{24}{7}a^{2}+\frac{31}{7}a+\frac{18}{7}$, $\frac{2}{7}a^{7}-\frac{3}{7}a^{6}+\frac{4}{7}a^{5}+a^{4}-\frac{4}{7}a^{3}-\frac{19}{7}a^{2}-\frac{2}{7}a+\frac{2}{7}$ | 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: | \( 33.8904383984 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 33.8904383984 \cdot 1}{6\cdot\sqrt{203233536}}\cr\approx \mathstrut & 0.617515300997 \end{aligned}\]
Galois group
$C_2^3:S_4$ (as 8T41):
A solvable group of order 192 |
The 14 conjugacy class representatives for $V_4^2:(S_3\times C_2)$ |
Character table for $V_4^2:(S_3\times C_2)$ |
Intermediate fields
\(\Q(\sqrt{-3}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 8 sibling: | data not computed |
Degree 12 siblings: | data not computed |
Degree 16 siblings: | data not computed |
Degree 24 siblings: | data not computed |
Degree 32 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 | R | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.2.0.1}{2} }^{4}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.8.3 | $x^{8} + 2 x^{7} + 32 x^{6} + 116 x^{5} + 456 x^{4} + 696 x^{3} + 1152 x^{2} + 432 x + 1296$ | $2$ | $4$ | $8$ | $C_2^3: C_4$ | $[2, 2, 2]^{4}$ |
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.6.7.5 | $x^{6} + 6 x^{2} + 3$ | $6$ | $1$ | $7$ | $D_{6}$ | $[3/2]_{2}^{2}$ | |
\(11\) | 11.4.2.2 | $x^{4} - 77 x^{2} + 242$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ |
11.4.0.1 | $x^{4} + 8 x^{2} + 10 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.44.2t1.a.a | $1$ | $ 2^{2} \cdot 11 $ | \(\Q(\sqrt{11}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
* | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
1.132.2t1.a.a | $1$ | $ 2^{2} \cdot 3 \cdot 11 $ | \(\Q(\sqrt{-33}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.1188.6t3.e.a | $2$ | $ 2^{2} \cdot 3^{3} \cdot 11 $ | 6.2.62099136.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
2.1188.3t2.a.a | $2$ | $ 2^{2} \cdot 3^{3} \cdot 11 $ | 3.1.1188.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
3.627264.6t8.h.a | $3$ | $ 2^{6} \cdot 3^{4} \cdot 11^{2}$ | 4.2.4752.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
3.14256.6t11.h.a | $3$ | $ 2^{4} \cdot 3^{4} \cdot 11 $ | 6.2.248396544.1 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ | |
3.4752.4t5.a.a | $3$ | $ 2^{4} \cdot 3^{3} \cdot 11 $ | 4.2.4752.1 | $S_4$ (as 4T5) | $1$ | $1$ | |
3.209088.6t11.g.a | $3$ | $ 2^{6} \cdot 3^{3} \cdot 11^{2}$ | 6.2.248396544.1 | $S_4\times C_2$ (as 6T11) | $1$ | $1$ | |
6.35769102336.12t108.a.a | $6$ | $ 2^{12} \cdot 3^{8} \cdot 11^{3}$ | 8.0.203233536.1 | $V_4^2:(S_3\times C_2)$ (as 8T41) | $1$ | $-2$ | |
* | 6.67744512.8t41.a.a | $6$ | $ 2^{8} \cdot 3^{7} \cdot 11^{2}$ | 8.0.203233536.1 | $V_4^2:(S_3\times C_2)$ (as 8T41) | $1$ | $0$ |
6.131153375232.12t108.a.a | $6$ | $ 2^{12} \cdot 3^{7} \cdot 11^{4}$ | 8.0.203233536.1 | $V_4^2:(S_3\times C_2)$ (as 8T41) | $1$ | $0$ | |
6.3974344704.8t41.a.a | $6$ | $ 2^{12} \cdot 3^{6} \cdot 11^{3}$ | 8.0.203233536.1 | $V_4^2:(S_3\times C_2)$ (as 8T41) | $1$ | $2$ |