Normalized defining polynomial
\( x^{10} - 3x^{9} + 5x^{8} - x^{7} - 8x^{6} + 27x^{5} - 55x^{4} + 18x^{3} - 39x^{2} + 2x - 7 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(30176086429637\) \(\medspace = 163^{3}\cdot 191^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(22.28\) | 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))
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Galois root discriminant: | $163^{1/2}191^{1/2}\approx 176.44545899512406$ | ||
Ramified primes: | \(163\), \(191\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{31133}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7}a^{7}+\frac{1}{7}a^{6}+\frac{3}{7}a^{5}+\frac{3}{7}a^{4}-\frac{2}{7}a^{3}+\frac{2}{7}a^{2}+\frac{1}{7}a$, $\frac{1}{7}a^{8}+\frac{2}{7}a^{6}+\frac{2}{7}a^{4}-\frac{3}{7}a^{3}-\frac{1}{7}a^{2}-\frac{1}{7}a$, $\frac{1}{3101}a^{9}+\frac{130}{3101}a^{8}+\frac{18}{3101}a^{7}+\frac{178}{3101}a^{6}-\frac{699}{3101}a^{5}-\frac{353}{3101}a^{4}-\frac{1375}{3101}a^{3}+\frac{545}{3101}a^{2}-\frac{1535}{3101}a+\frac{10}{443}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
Unit group
Rank: | $5$ | 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)
| |
Fundamental units: | $\frac{899}{3101}a^{9}-\frac{1854}{3101}a^{8}+\frac{1563}{3101}a^{7}+\frac{4086}{3101}a^{6}-\frac{8644}{3101}a^{5}+\frac{15346}{3101}a^{4}-\frac{22748}{3101}a^{3}-\frac{37658}{3101}a^{2}-\frac{10652}{3101}a-\frac{2085}{443}$, $\frac{547}{3101}a^{9}-\frac{1542}{3101}a^{8}+\frac{2315}{3101}a^{7}+\frac{349}{3101}a^{6}-\frac{4917}{3101}a^{5}+\frac{14233}{3101}a^{4}-\frac{26048}{3101}a^{3}+\frac{313}{443}a^{2}-\frac{11678}{3101}a-\frac{289}{443}$, $\frac{1035}{3101}a^{9}-\frac{3223}{3101}a^{8}+\frac{4897}{3101}a^{7}+\frac{55}{443}a^{6}-\frac{11121}{3101}a^{5}+\frac{28029}{3101}a^{4}-\frac{52040}{3101}a^{3}+\frac{7667}{3101}a^{2}-\frac{13417}{3101}a+\frac{161}{443}$, $\frac{995}{3101}a^{9}-\frac{2664}{3101}a^{8}+\frac{3734}{3101}a^{7}+\frac{177}{443}a^{6}-\frac{9298}{3101}a^{5}+\frac{24429}{3101}a^{4}-\frac{44441}{3101}a^{3}-\frac{8374}{3101}a^{2}-\frac{17138}{3101}a-\frac{1125}{443}$, $\frac{2252}{3101}a^{9}-\frac{895}{443}a^{8}+\frac{8640}{3101}a^{7}+\frac{3485}{3101}a^{6}-\frac{3315}{443}a^{5}+\frac{55604}{3101}a^{4}-\frac{98276}{3101}a^{3}-\frac{2372}{443}a^{2}-\frac{26671}{3101}a-\frac{959}{443}$ | 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: | \( 822.418264941 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 822.418264941 \cdot 2}{2\cdot\sqrt{30176086429637}}\cr\approx \mathstrut & 0.933341881524 \end{aligned}\]
Galois group
A non-solvable group of order 720 |
The 11 conjugacy class representatives for $S_{6}$ |
Character table for $S_{6}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 6 siblings: | 6.2.31133.1, 6.2.30176086429637.1 |
Degree 12 siblings: | data not computed |
Degree 15 siblings: | data not computed |
Degree 20 siblings: | data not computed |
Degree 30 siblings: | data not computed |
Degree 36 sibling: | data not computed |
Degree 40 siblings: | data not computed |
Degree 45 sibling: | data not computed |
Minimal sibling: | 6.2.31133.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.3.0.1}{3} }^{3}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.5.0.1}{5} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(163\) | 163.2.1.2 | $x^{2} + 163$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
163.4.0.1 | $x^{4} + 8 x^{2} + 91 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
163.4.2.2 | $x^{4} - 25917 x^{2} + 53138$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
\(191\) | $\Q_{191}$ | $x + 172$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
191.3.0.1 | $x^{3} + 4 x + 172$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
191.6.3.1 | $x^{6} + 145924 x^{2} - 1198473812$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |