Normalized defining polynomial
\( x^{10} - 2x^{9} - 2x^{8} + 3x^{7} + 6x^{6} - 13x^{5} - 11x^{4} + 12x^{3} + x^{2} - 3x + 1 \)
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: | \(-1541199117151\) \(\medspace = -\,11551^{3}\) | 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: | \(16.55\) | 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: | $11551^{1/2}\approx 107.47557862137798$ | ||
Ramified primes: | \(11551\) | 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: | \(\Q(\sqrt{-11551}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{709}a^{9}-\frac{219}{709}a^{8}+\frac{18}{709}a^{7}+\frac{351}{709}a^{6}-\frac{298}{709}a^{5}+\frac{134}{709}a^{4}-\frac{20}{709}a^{3}+\frac{98}{709}a^{2}+\frac{5}{709}a+\frac{330}{709}$
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: | $a^{9}-2a^{8}-2a^{7}+3a^{6}+6a^{5}-13a^{4}-11a^{3}+12a^{2}+a-3$, $\frac{577}{709}a^{9}-\frac{870}{709}a^{8}-\frac{1667}{709}a^{7}+\frac{1171}{709}a^{6}+\frac{3886}{709}a^{5}-\frac{5635}{709}a^{4}-\frac{9413}{709}a^{3}+\frac{3371}{709}a^{2}+\frac{2176}{709}a-\frac{1020}{709}$, $\frac{132}{709}a^{9}-\frac{548}{709}a^{8}+\frac{249}{709}a^{7}+\frac{956}{709}a^{6}+\frac{368}{709}a^{5}-\frac{3582}{709}a^{4}+\frac{1614}{709}a^{3}+\frac{5137}{709}a^{2}-\frac{1467}{709}a-\frac{1107}{709}$, $\frac{125}{709}a^{9}-\frac{433}{709}a^{8}+\frac{123}{709}a^{7}+\frac{626}{709}a^{6}+\frac{327}{709}a^{5}-\frac{2393}{709}a^{4}+\frac{1045}{709}a^{3}+\frac{2324}{709}a^{2}-\frac{793}{709}a+\frac{128}{709}$, $\frac{132}{709}a^{9}-\frac{548}{709}a^{8}+\frac{249}{709}a^{7}+\frac{956}{709}a^{6}+\frac{368}{709}a^{5}-\frac{3582}{709}a^{4}+\frac{1614}{709}a^{3}+\frac{5137}{709}a^{2}-\frac{1467}{709}a-\frac{398}{709}$, $\frac{705}{709}a^{9}-\frac{1251}{709}a^{8}-\frac{1490}{709}a^{7}+\frac{1432}{709}a^{6}+\frac{4028}{709}a^{5}-\frac{7626}{709}a^{4}-\frac{8428}{709}a^{3}+\frac{4571}{709}a^{2}-\frac{1438}{709}a+\frac{98}{709}$ | 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: | \( 166.027443935 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 166.027443935 \cdot 1}{2\cdot\sqrt{1541199117151}}\cr\approx \mathstrut & 0.265387276806 \end{aligned}\]
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: | 5.3.11551.1 |
Degree 6 sibling: | 6.0.1541199117151.2 |
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: | 5.3.11551.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.3.0.1}{3} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(11551\) | $\Q_{11551}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $6$ | $2$ | $3$ | $3$ |