Normalized defining polynomial
\( x^{10} - x^{9} - x^{8} - 2x^{7} + x^{6} - 2x^{4} + x^{3} + x^{2} + 2x - 1 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[4, 3]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-3196989667\) \(\medspace = -\,29\cdot 73^{2}\cdot 137\cdot 151\) | 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: | \(8.92\) | 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: | $29^{1/2}73^{2/3}137^{1/2}151^{1/2}\approx 13528.915874373493$ | ||
Ramified primes: | \(29\), \(73\), \(137\), \(151\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | \(\Q(\sqrt{-599923}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{71}a^{9}+\frac{19}{71}a^{8}+\frac{24}{71}a^{7}-\frac{19}{71}a^{6}-\frac{24}{71}a^{5}+\frac{17}{71}a^{4}-\frac{17}{71}a^{3}+\frac{16}{71}a^{2}-\frac{34}{71}a+\frac{32}{71}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | 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: | $\frac{25}{71}a^{9}-\frac{22}{71}a^{8}-\frac{39}{71}a^{7}-\frac{49}{71}a^{6}+\frac{39}{71}a^{5}-\frac{1}{71}a^{4}+\frac{1}{71}a^{3}+\frac{45}{71}a^{2}+\frac{73}{71}a+\frac{19}{71}$, $a^{9}-a^{8}-a^{7}-2a^{6}+a^{5}-2a^{3}+a^{2}+a+2$, $\frac{9}{71}a^{9}-\frac{42}{71}a^{8}+\frac{3}{71}a^{7}+\frac{42}{71}a^{6}+\frac{68}{71}a^{5}+\frac{11}{71}a^{4}-\frac{11}{71}a^{3}+\frac{73}{71}a^{2}+\frac{49}{71}a+\frac{4}{71}$, $\frac{67}{71}a^{9}-\frac{76}{71}a^{8}-\frac{25}{71}a^{7}-\frac{137}{71}a^{6}+\frac{25}{71}a^{5}-\frac{68}{71}a^{4}-\frac{145}{71}a^{3}+\frac{78}{71}a^{2}-\frac{6}{71}a+\frac{85}{71}$, $\frac{36}{71}a^{9}-\frac{26}{71}a^{8}-\frac{59}{71}a^{7}-\frac{45}{71}a^{6}-\frac{12}{71}a^{5}+\frac{44}{71}a^{4}-\frac{115}{71}a^{3}+\frac{8}{71}a^{2}+\frac{54}{71}a+\frac{16}{71}$, $\frac{86}{71}a^{9}-\frac{70}{71}a^{8}-\frac{66}{71}a^{7}-\frac{214}{71}a^{6}-\frac{5}{71}a^{5}-\frac{29}{71}a^{4}-\frac{184}{71}a^{3}+\frac{27}{71}a^{2}+\frac{58}{71}a+\frac{196}{71}$ | 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: | \( 4.898963260479714 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 4.898963260479714 \cdot 1}{2\cdot\sqrt{3196989667}}\cr\approx \mathstrut & 0.171934550777679 \end{aligned}\]
Galois group
A non-solvable group of order 3628800 |
The 42 conjugacy class representatives for $S_{10}$ |
Character table for $S_{10}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 20 sibling: | data not computed |
Degree 45 sibling: | data not computed |
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 | ${\href{/padicField/2.10.0.1}{10} }$ | ${\href{/padicField/3.10.0.1}{10} }$ | ${\href{/padicField/5.10.0.1}{10} }$ | ${\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.10.0.1}{10} }$ | ${\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ | ${\href{/padicField/17.10.0.1}{10} }$ | ${\href{/padicField/19.10.0.1}{10} }$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
---|---|---|---|---|---|---|---|
\(29\) | 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
29.4.0.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(73\) | $\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
73.2.0.1 | $x^{2} + 70 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
73.3.2.1 | $x^{3} + 73$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
73.4.0.1 | $x^{4} + 16 x^{2} + 56 x + 5$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(137\) | 137.2.0.1 | $x^{2} + 131 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
137.2.1.1 | $x^{2} + 137$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
137.2.0.1 | $x^{2} + 131 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
137.4.0.1 | $x^{4} + x^{2} + 95 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
\(151\) | 151.2.0.1 | $x^{2} + 149 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
151.2.1.2 | $x^{2} + 151$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
151.6.0.1 | $x^{6} + 125 x^{3} + 18 x^{2} + 15 x + 6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |