Normalized defining polynomial
\( x^{8} - 3x^{7} + 7x^{6} + 5x^{5} - 31x^{4} + 70x^{3} - 68x^{2} + 43x - 11 \)
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: | \(-16836267547\) \(\medspace = -\,11^{3}\cdot 233^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(18.98\) | 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: | $11^{1/2}233^{1/2}\approx 50.62608023538856$ | ||
Ramified primes: | \(11\), \(233\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-2563}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{1573}a^{7}-\frac{42}{143}a^{6}-\frac{290}{1573}a^{5}-\frac{590}{1573}a^{4}+\frac{223}{1573}a^{3}-\frac{42}{1573}a^{2}+\frac{334}{1573}a-\frac{62}{143}$
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: | $\frac{61}{1573}a^{7}+\frac{12}{143}a^{6}-\frac{387}{1573}a^{5}+\frac{1762}{1573}a^{4}+\frac{1019}{1573}a^{3}-\frac{5708}{1573}a^{2}+\frac{9363}{1573}a-\frac{493}{143}$, $\frac{373}{1573}a^{7}-\frac{79}{143}a^{6}+\frac{1940}{1573}a^{5}+\frac{3296}{1573}a^{4}-\frac{9628}{1573}a^{3}+\frac{18940}{1573}a^{2}-\frac{12269}{1573}a+\frac{612}{143}$, $\frac{323}{1573}a^{7}-\frac{124}{143}a^{6}+\frac{2283}{1573}a^{5}+\frac{1336}{1573}a^{4}-\frac{16059}{1573}a^{3}+\frac{22613}{1573}a^{2}-\frac{17958}{1573}a+\frac{566}{143}$, $\frac{3664}{1573}a^{7}-\frac{878}{143}a^{6}+\frac{21237}{1573}a^{5}+\frac{27856}{1573}a^{4}-\frac{106279}{1573}a^{3}+\frac{207902}{1573}a^{2}-\frac{151026}{1573}a+\frac{7495}{143}$ | 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: | \( 114.24970385 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 114.24970385 \cdot 1}{2\cdot\sqrt{16836267547}}\cr\approx \mathstrut & 0.43681925564 \end{aligned}\]
Galois group
$\GL(2,3)$ (as 8T23):
A solvable group of order 48 |
The 8 conjugacy class representatives for $\textrm{GL(2,3)}$ |
Character table for $\textrm{GL(2,3)}$ |
Intermediate fields
4.2.2563.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 16 sibling: | deg 16 |
Degree 24 sibling: | deg 24 |
Arithmetically equvalently sibling: | 8.2.16836267547.3 |
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 | ${\href{/padicField/2.8.0.1}{8} }$ | ${\href{/padicField/3.8.0.1}{8} }$ | ${\href{/padicField/5.8.0.1}{8} }$ | ${\href{/padicField/7.2.0.1}{2} }^{3}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.8.0.1}{8} }$ | ${\href{/padicField/53.8.0.1}{8} }$ | ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 | $[\ ]$ | |
11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
11.2.1.1 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(233\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
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 | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.2563.2t1.a.a | $1$ | $ 11 \cdot 233 $ | \(\Q(\sqrt{-2563}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
2.2563.3t2.a.a | $2$ | $ 11 \cdot 233 $ | 3.1.2563.1 | $S_3$ (as 3T2) | $1$ | $0$ | |
2.2563.24t22.b.a | $2$ | $ 11 \cdot 233 $ | 8.2.16836267547.2 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
2.2563.24t22.b.b | $2$ | $ 11 \cdot 233 $ | 8.2.16836267547.2 | $\textrm{GL(2,3)}$ (as 8T23) | $0$ | $0$ | |
3.6568969.6t8.a.a | $3$ | $ 11^{2} \cdot 233^{2}$ | 4.2.2563.1 | $S_4$ (as 4T5) | $1$ | $-1$ | |
* | 3.2563.4t5.a.a | $3$ | $ 11 \cdot 233 $ | 4.2.2563.1 | $S_4$ (as 4T5) | $1$ | $1$ |
* | 4.6568969.8t23.b.a | $4$ | $ 11^{2} \cdot 233^{2}$ | 8.2.16836267547.2 | $\textrm{GL(2,3)}$ (as 8T23) | $1$ | $0$ |