Normalized defining polynomial
\( x^{8} - 56x^{5} - 1050x^{4} - 8400x^{3} - 35000x^{2} - 75000x - 65625 \)
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)
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Discriminant: | \(-1316376608184278779428864\) \(\medspace = -\,2^{29}\cdot 3^{11}\cdot 7^{12}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(1034.96\) | 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: | $2^{137/32}3^{37/18}7^{12/7}\approx 5227.251692600439$ | ||
Ramified primes: | \(2\), \(3\), \(7\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-6}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5}a^{4}-\frac{1}{5}a$, $\frac{1}{75}a^{5}-\frac{31}{75}a^{2}$, $\frac{1}{375}a^{6}-\frac{181}{375}a^{3}+\frac{1}{5}a^{2}-\frac{2}{5}a$, $\frac{1}{1875}a^{7}-\frac{181}{1875}a^{4}+\frac{11}{25}a^{3}-\frac{12}{25}a^{2}+\frac{2}{5}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$ (assuming GRH)
Unit group
Rank: | $4$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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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)
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Fundamental units: | $\frac{13\!\cdots\!47}{1875}a^{7}-\frac{89\!\cdots\!58}{375}a^{6}-\frac{37\!\cdots\!53}{15}a^{5}-\frac{15\!\cdots\!57}{1875}a^{4}-\frac{28\!\cdots\!47}{375}a^{3}-\frac{26\!\cdots\!72}{75}a^{2}-\frac{27\!\cdots\!54}{5}a-22\!\cdots\!81$, $\frac{11\!\cdots\!47}{1875}a^{7}-\frac{15\!\cdots\!94}{75}a^{6}+\frac{27\!\cdots\!22}{5}a^{5}-\frac{83\!\cdots\!82}{1875}a^{4}-\frac{72\!\cdots\!92}{15}a^{3}-\frac{77\!\cdots\!99}{25}a^{2}-\frac{43\!\cdots\!93}{5}a-70\!\cdots\!06$, $\frac{26\!\cdots\!38}{1875}a^{7}-\frac{10\!\cdots\!34}{25}a^{6}-\frac{10\!\cdots\!99}{75}a^{5}-\frac{28\!\cdots\!03}{1875}a^{4}-\frac{36\!\cdots\!53}{25}a^{3}-\frac{52\!\cdots\!49}{75}a^{2}-\frac{58\!\cdots\!54}{5}a-55\!\cdots\!11$, $\frac{47\!\cdots\!38}{1875}a^{7}-\frac{47\!\cdots\!07}{375}a^{6}-\frac{42\!\cdots\!62}{25}a^{5}-\frac{14\!\cdots\!78}{1875}a^{4}-\frac{96\!\cdots\!88}{375}a^{3}-\frac{48\!\cdots\!44}{25}a^{2}-\frac{30\!\cdots\!34}{5}a-66\!\cdots\!00$ (assuming GRH) | 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: | \( 1754817880.49 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 1754817880.49 \cdot 2}{2\cdot\sqrt{1316376608184278779428864}}\cr\approx \mathstrut & 1.51754496867 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 40320 |
The 22 conjugacy class representatives for $S_8$ |
Character table for $S_8$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 16 sibling: | deg 16 |
Degree 28 sibling: | deg 28 |
Degree 30 sibling: | deg 30 |
Degree 35 sibling: | deg 35 |
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 | ${\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.8.0.1}{8} }$ | ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.29.38 | $x^{8} + 20 x^{6} + 16 x^{5} + 16 x^{3} + 10$ | $8$ | $1$ | $29$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $[2, 3, 7/2, 4, 17/4, 19/4]$ |
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.6.10.5 | $x^{6} - 12 x^{3} + 360$ | $3$ | $2$ | $10$ | $S_3^2$ | $[3/2, 5/2]_{2}^{2}$ | |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.7.12.1 | $x^{7} + 42 x^{6} + 7$ | $7$ | $1$ | $12$ | $C_7$ | $[2]$ |