Normalized defining polynomial
\( x^{8} - 3528x^{2} - 6048x - 2646 \)
Invariants
Degree: | $8$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-65006252256013766885376\) \(\medspace = -\,2^{31}\cdot 3^{7}\cdot 7^{12}\) | 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: | \(710.59\) | 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: | $2^{141/32}3^{7/6}7^{12/7}\approx 2146.813805316912$ | ||
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)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}$, $\frac{1}{3}a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{45}a^{6}+\frac{1}{15}a^{5}+\frac{1}{15}a^{4}+\frac{2}{15}a^{3}+\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{1575}a^{7}+\frac{2}{225}a^{6}+\frac{1}{75}a^{5}-\frac{11}{75}a^{4}-\frac{4}{75}a^{3}-\frac{2}{25}a^{2}-\frac{9}{25}a+\frac{3}{25}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
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: | $\frac{50\!\cdots\!03}{75}a^{7}-\frac{12\!\cdots\!14}{225}a^{6}+\frac{31\!\cdots\!98}{75}a^{5}-\frac{23\!\cdots\!83}{75}a^{4}+\frac{16\!\cdots\!68}{75}a^{3}-\frac{37\!\cdots\!26}{25}a^{2}-\frac{59\!\cdots\!32}{25}a-\frac{54\!\cdots\!51}{25}$, $\frac{27\!\cdots\!61}{525}a^{7}-\frac{14\!\cdots\!98}{75}a^{6}-\frac{10\!\cdots\!02}{75}a^{5}+\frac{23\!\cdots\!27}{75}a^{4}-\frac{21\!\cdots\!59}{25}a^{3}-\frac{44\!\cdots\!16}{25}a^{2}-\frac{97\!\cdots\!32}{25}a+\frac{69\!\cdots\!89}{25}$, $\frac{11\!\cdots\!23}{525}a^{7}+\frac{41\!\cdots\!78}{225}a^{6}+\frac{34\!\cdots\!34}{75}a^{5}-\frac{24\!\cdots\!23}{25}a^{4}-\frac{29\!\cdots\!57}{25}a^{3}-\frac{91\!\cdots\!38}{25}a^{2}-\frac{99\!\cdots\!56}{25}a-\frac{35\!\cdots\!03}{25}$, $\frac{61\!\cdots\!24}{225}a^{7}+\frac{25\!\cdots\!46}{225}a^{6}+\frac{35\!\cdots\!78}{75}a^{5}+\frac{14\!\cdots\!12}{75}a^{4}+\frac{20\!\cdots\!66}{25}a^{3}+\frac{84\!\cdots\!14}{25}a^{2}+\frac{11\!\cdots\!48}{25}a+\frac{43\!\cdots\!89}{25}$ (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)]
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Regulator: | \( 683277868.793 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{3}\cdot 683277868.793 \cdot 1}{2\cdot\sqrt{65006252256013766885376}}\cr\approx \mathstrut & 1.32950321108 \end{aligned}\] (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 | ${\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.4.0.1}{4} }^{2}$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.8.0.1}{8} }$ | ${\href{/padicField/41.8.0.1}{8} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.7.0.1}{7} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.8.31.153 | $x^{8} + 16 x^{7} + 16 x^{5} + 8 x^{2} + 42$ | $8$ | $1$ | $31$ | $C_2 \wr C_2\wr C_2$ | $[2, 3, 7/2, 4, 17/4, 5]^{2}$ |
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.6.6.4 | $x^{6} + 48 x^{4} + 6 x^{3} + 36 x^{2} + 36 x + 9$ | $3$ | $2$ | $6$ | $D_{6}$ | $[3/2]_{2}^{2}$ | |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.7.12.1 | $x^{7} + 42 x^{6} + 7$ | $7$ | $1$ | $12$ | $C_7$ | $[2]$ |