Normalized defining polynomial
\( x^{10} - x^{9} + x^{8} + 4x^{7} - 15x^{6} + x^{5} + 27x^{4} - 14x^{3} - 13x^{2} + 11x - 1 \)
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: | \(107365139712\) \(\medspace = 2^{8}\cdot 3^{4}\cdot 173^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(12.68\) | 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: | $2^{4/5}3^{1/2}173^{1/2}\approx 39.66501980382454$ | ||
Ramified primes: | \(2\), \(3\), \(173\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{173}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{317}a^{9}+\frac{93}{317}a^{8}-\frac{133}{317}a^{7}-\frac{135}{317}a^{6}-\frac{25}{317}a^{5}-\frac{130}{317}a^{4}-\frac{147}{317}a^{3}+\frac{116}{317}a^{2}+\frac{113}{317}a-\frac{145}{317}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | 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: | $\frac{248}{317}a^{9}-\frac{77}{317}a^{8}+\frac{301}{317}a^{7}+\frac{1073}{317}a^{6}-\frac{2713}{317}a^{5}-\frac{1491}{317}a^{4}+\frac{4437}{317}a^{3}-\frac{79}{317}a^{2}-\frac{2408}{317}a+\frac{812}{317}$, $\frac{77}{317}a^{9}-\frac{130}{317}a^{8}+\frac{220}{317}a^{7}+\frac{66}{317}a^{6}-\frac{974}{317}a^{5}+\frac{768}{317}a^{4}+\frac{1044}{317}a^{3}-\frac{895}{317}a^{2}-\frac{175}{317}a+\frac{247}{317}$, $\frac{218}{317}a^{9}-\frac{14}{317}a^{8}+\frac{170}{317}a^{7}+\frac{1002}{317}a^{6}-\frac{2280}{317}a^{5}-\frac{2346}{317}a^{4}+\frac{4409}{317}a^{3}+\frac{1196}{317}a^{2}-\frac{2628}{317}a+\frac{724}{317}$, $\frac{45}{317}a^{9}+\frac{64}{317}a^{8}+\frac{38}{317}a^{7}+\frac{265}{317}a^{6}-\frac{174}{317}a^{5}-\frac{1095}{317}a^{4}+\frac{42}{317}a^{3}+\frac{1416}{317}a^{2}+\frac{13}{317}a-\frac{819}{317}$, $\frac{251}{317}a^{9}-\frac{115}{317}a^{8}+\frac{219}{317}a^{7}+\frac{985}{317}a^{6}-\frac{3105}{317}a^{5}-\frac{1564}{317}a^{4}+\frac{5264}{317}a^{3}+\frac{586}{317}a^{2}-\frac{2386}{317}a+\frac{377}{317}$ | 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: | \( 40.8746207381 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 40.8746207381 \cdot 1}{2\cdot\sqrt{107365139712}}\cr\approx \mathstrut & 0.388840558569 \end{aligned}\]
Galois group
A non-solvable group of order 120 |
The 7 conjugacy class representatives for $S_5$ |
Character table for $S_5$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 5 sibling: | 5.1.24912.1 |
Degree 6 sibling: | 6.2.745591248.1 |
Degree 10 sibling: | data not computed |
Degree 12 sibling: | data not computed |
Degree 15 sibling: | data not computed |
Degree 20 siblings: | data not computed |
Degree 24 sibling: | data not computed |
Degree 30 siblings: | data not computed |
Degree 40 sibling: | data not computed |
Minimal sibling: | 5.1.24912.1 |
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 | ${\href{/padicField/5.4.0.1}{4} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.5.0.1}{5} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.10.8.1 | $x^{10} + 5 x^{9} + 15 x^{8} + 30 x^{7} + 45 x^{6} + 55 x^{5} + 55 x^{4} + 10 x^{3} - 25 x^{2} - 5 x + 7$ | $5$ | $2$ | $8$ | $F_5$ | $[\ ]_{5}^{4}$ |
\(3\) | 3.2.0.1 | $x^{2} + 2 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
3.4.2.2 | $x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
\(173\) | $\Q_{173}$ | $x + 171$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
173.3.0.1 | $x^{3} + 2 x + 171$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
173.6.3.1 | $x^{6} + 83040 x^{5} + 2298547723 x^{4} + 21207957785622 x^{3} + 402260049631 x^{2} + 42811394473266 x + 3626818487364574$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |