Normalized defining polynomial
\( x^{8} + 14x^{6} - 54x^{5} - 98x^{4} - 378x^{3} + 799x^{2} + 2870x + 5677 \)
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: | \(1458783255601\) \(\medspace = 7^{4}\cdot 157^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(33.15\) | 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: | $7^{1/2}157^{1/2}\approx 33.15116890850155$ | ||
Ramified primes: | \(7\), \(157\) | 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 not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{18}a^{6}-\frac{2}{9}a^{5}+\frac{1}{18}a^{4}-\frac{5}{18}a^{3}+\frac{1}{18}a^{2}-\frac{1}{6}a-\frac{1}{18}$, $\frac{1}{868675338}a^{7}+\frac{1125043}{144779223}a^{6}-\frac{14656675}{289558446}a^{5}-\frac{71788405}{868675338}a^{4}-\frac{8576291}{29954322}a^{3}+\frac{106796617}{868675338}a^{2}-\frac{354494317}{868675338}a-\frac{8794301}{434337669}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{6}$, which has order $12$
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: | $\frac{323149}{868675338}a^{7}-\frac{585379}{434337669}a^{6}+\frac{2317600}{434337669}a^{5}-\frac{6427978}{144779223}a^{4}+\frac{202231}{9984774}a^{3}-\frac{22349141}{434337669}a^{2}+\frac{628362287}{868675338}a+\frac{534445987}{868675338}$, $\frac{176177}{434337669}a^{7}-\frac{927853}{868675338}a^{6}+\frac{1751672}{434337669}a^{5}-\frac{22747889}{868675338}a^{4}-\frac{252043}{29954322}a^{3}+\frac{6653285}{289558446}a^{2}+\frac{349733363}{868675338}a-\frac{26238675}{96519482}$, $\frac{194387753}{868675338}a^{7}-\frac{539856649}{868675338}a^{6}-\frac{1320570031}{434337669}a^{5}-\frac{15357212293}{289558446}a^{4}-\frac{1065873209}{4992387}a^{3}-\frac{590947556201}{868675338}a^{2}-\frac{492427487032}{434337669}a-\frac{488379809501}{434337669}$ | 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: | \( 81.8345338489 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{4}\cdot 81.8345338489 \cdot 12}{2\cdot\sqrt{1458783255601}}\cr\approx \mathstrut & 0.633595307101 \end{aligned}\]
Galois group
A solvable group of order 24 |
The 5 conjugacy class representatives for $S_4$ |
Character table for $S_4$ |
Intermediate fields
\(\Q(\sqrt{-1099}) \), 4.2.1099.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | deg 24 |
Degree 4 sibling: | 4.2.1099.1 |
Degree 6 siblings: | 6.2.1207801.1, 6.0.1327373299.4 |
Degree 12 siblings: | deg 12, deg 12 |
Minimal sibling: | 4.2.1099.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}$ | ${\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{4}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | 7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(157\) | 157.2.1.2 | $x^{2} + 314$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
157.2.1.2 | $x^{2} + 314$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
157.2.1.2 | $x^{2} + 314$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
157.2.1.2 | $x^{2} + 314$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |