Normalized defining polynomial
\( x^{8} - x^{7} - 10x^{6} + 32x^{5} + 43x^{4} - 245x^{3} + 393x^{2} - 192x + 68 \)
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: | \(529414856881\) \(\medspace = 853^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(29.21\) | 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: | $853^{2/3}\approx 89.94272075480275$ | ||
Ramified primes: | \(853\) | 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 a CM field. | |||
Reflex fields: | 8.0.529414856881.1$^{8}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3657616}a^{7}-\frac{1371723}{3657616}a^{6}-\frac{339511}{914404}a^{5}-\frac{196645}{457202}a^{4}-\frac{394197}{3657616}a^{3}+\frac{1378013}{3657616}a^{2}+\frac{1543191}{3657616}a+\frac{792221}{1828808}$
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: | \( -1 \) (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: | $\frac{3891}{228601}a^{7}+\frac{1955}{228601}a^{6}-\frac{37089}{228601}a^{5}+\frac{83417}{228601}a^{4}+\frac{320784}{228601}a^{3}-\frac{673675}{228601}a^{2}+\frac{350916}{228601}a+\frac{837857}{228601}$, $\frac{15596}{228601}a^{7}+\frac{4076}{228601}a^{6}-\frac{171574}{228601}a^{5}+\frac{284768}{228601}a^{4}+\frac{1241887}{228601}a^{3}-\frac{2718277}{228601}a^{2}+\frac{1407960}{228601}a+\frac{2618347}{228601}$, $\frac{19905}{1828808}a^{7}-\frac{42875}{1828808}a^{6}-\frac{63693}{457202}a^{5}+\frac{116198}{228601}a^{4}+\frac{923843}{1828808}a^{3}-\frac{6428851}{1828808}a^{2}+\frac{9701727}{1828808}a-\frac{1566783}{914404}$ | 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: | \( 282.299934222 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 282.299934222 \cdot 1}{2\cdot\sqrt{529414856881}}\cr\approx \mathstrut & 0.302344583330 \end{aligned}\]
Galois group
$\SL(2,3)$ (as 8T12):
A solvable group of order 24 |
The 7 conjugacy class representatives for $\SL(2,3)$ |
Character table for $\SL(2,3)$ |
Intermediate fields
4.4.727609.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
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 | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.3.0.1}{3} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(853\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $6$ | $3$ | $2$ | $4$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
1.853.3t1.a.a | $1$ | $ 853 $ | 3.3.727609.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
1.853.3t1.a.b | $1$ | $ 853 $ | 3.3.727609.1 | $C_3$ (as 3T1) | $0$ | $1$ | |
2.727609.24t7.a.a | $2$ | $ 853^{2}$ | 8.0.529414856881.1 | $\SL(2,3)$ (as 8T12) | $-1$ | $-2$ | |
* | 2.853.8t12.a.a | $2$ | $ 853 $ | 8.0.529414856881.1 | $\SL(2,3)$ (as 8T12) | $0$ | $-2$ |
* | 2.853.8t12.a.b | $2$ | $ 853 $ | 8.0.529414856881.1 | $\SL(2,3)$ (as 8T12) | $0$ | $-2$ |
* | 3.727609.4t4.a.a | $3$ | $ 853^{2}$ | 4.4.727609.1 | $A_4$ (as 4T4) | $1$ | $3$ |