Normalized defining polynomial
\( x^{8} - 2x^{7} + 9x^{6} - 43x^{5} + 84x^{4} - 173x^{3} + 218x^{2} - 137x + 52 \)
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: | \(-6626684877131\) \(\medspace = -\,11\cdot 881^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(40.06\) | 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: | $11^{1/2}881^{1/2}\approx 98.44287683728061$ | ||
Ramified primes: | \(11\), \(881\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-11}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{112011}a^{7}-\frac{21706}{112011}a^{6}-\frac{11233}{112011}a^{5}-\frac{46958}{112011}a^{4}-\frac{11573}{112011}a^{3}+\frac{51557}{112011}a^{2}-\frac{3020}{112011}a+\frac{19508}{112011}$
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: | $\frac{2621}{112011}a^{7}+\frac{10162}{112011}a^{6}+\frac{17200}{112011}a^{5}+\frac{23171}{112011}a^{4}-\frac{201874}{112011}a^{3}+\frac{269653}{112011}a^{2}-\frac{858727}{112011}a+\frac{837529}{112011}$, $\frac{1564}{112011}a^{7}-\frac{8851}{112011}a^{6}+\frac{17315}{112011}a^{5}-\frac{75107}{112011}a^{4}+\frac{269632}{112011}a^{3}-\frac{348805}{112011}a^{2}+\frac{205204}{112011}a-\frac{68491}{112011}$, $\frac{10360}{112011}a^{7}+\frac{43928}{112011}a^{6}+\frac{117560}{112011}a^{5}+\frac{90904}{112011}a^{4}-\frac{380543}{112011}a^{3}+\frac{174083}{112011}a^{2}-\frac{36131}{112011}a-\frac{76975}{112011}$, $\frac{160189}{112011}a^{7}+\frac{95039}{112011}a^{6}+\frac{1061777}{112011}a^{5}-\frac{5768918}{112011}a^{4}+\frac{7979545}{112011}a^{3}-\frac{24125155}{112011}a^{2}+\frac{16246324}{112011}a-\frac{9436801}{112011}$ | 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: | \( 6614.27055136 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 6614.27055136 \cdot 1}{2\cdot\sqrt{6626684877131}}\cr\approx \mathstrut & 1.27468653067 \end{aligned}\]
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
\(\Q(\sqrt{881}) \) |
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 | ${\href{/padicField/2.4.0.1}{4} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.8.0.1}{8} }$ | R | ${\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(11\) | $\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{11}$ | $x + 9$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
11.2.1.2 | $x^{2} + 11$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.0.1 | $x^{2} + 7 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
\(881\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.11.2t1.a.a | $1$ | $ 11 $ | \(\Q(\sqrt{-11}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
1.9691.2t1.a.a | $1$ | $ 11 \cdot 881 $ | \(\Q(\sqrt{-9691}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
* | 1.881.2t1.a.a | $1$ | $ 881 $ | \(\Q(\sqrt{881}) \) | $C_2$ (as 2T1) | $1$ | $1$ |
2.9691.4t3.a.a | $2$ | $ 11 \cdot 881 $ | 4.0.106601.1 | $D_{4}$ (as 4T3) | $1$ | $0$ | |
4.82739538761.12t34.a.a | $4$ | $ 11^{2} \cdot 881^{3}$ | 6.0.1172611.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.1033070291.12t34.a.a | $4$ | $ 11^{3} \cdot 881^{2}$ | 6.0.1172611.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $-2$ | |
4.106601.6t13.a.a | $4$ | $ 11^{2} \cdot 881 $ | 6.0.1172611.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $0$ | |
4.8537771.6t13.a.a | $4$ | $ 11 \cdot 881^{2}$ | 6.0.1172611.1 | $C_3^2:D_4$ (as 6T13) | $1$ | $2$ | |
6.100...081.12t201.a.a | $6$ | $ 11^{4} \cdot 881^{3}$ | 8.2.6626684877131.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
6.110...891.12t202.a.a | $6$ | $ 11^{5} \cdot 881^{3}$ | 8.2.6626684877131.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ | |
* | 6.7521776251.8t47.a.a | $6$ | $ 11 \cdot 881^{3}$ | 8.2.6626684877131.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |
6.82739538761.12t200.a.a | $6$ | $ 11^{2} \cdot 881^{3}$ | 8.2.6626684877131.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
9.910134926371.16t1294.a.a | $9$ | $ 11^{3} \cdot 881^{3}$ | 8.2.6626684877131.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
9.121...801.18t272.a.a | $9$ | $ 11^{6} \cdot 881^{3}$ | 8.2.6626684877131.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.828...641.18t273.a.a | $9$ | $ 11^{6} \cdot 881^{6}$ | 8.2.6626684877131.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $1$ | |
9.622...011.18t274.a.a | $9$ | $ 11^{3} \cdot 881^{6}$ | 8.2.6626684877131.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-1$ | |
12.911...051.36t1763.a.a | $12$ | $ 11^{7} \cdot 881^{6}$ | 8.2.6626684877131.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $-2$ | |
12.753...331.24t2821.a.a | $12$ | $ 11^{5} \cdot 881^{6}$ | 8.2.6626684877131.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $2$ | |
18.753...811.36t1758.a.a | $18$ | $ 11^{9} \cdot 881^{9}$ | 8.2.6626684877131.1 | $S_4\wr C_2$ (as 8T47) | $1$ | $0$ |