Normalized defining polynomial
\( x^{10} - x^{7} - x^{6} - 4x^{5} - 21x^{4} - 10x^{3} - x^{2} + 2x + 1 \)
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: | \(-12712961507219\) \(\medspace = -\,23339^{3}\) | 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: | \(20.44\) | 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: | $23339^{1/2}\approx 152.77107055984126$ | ||
Ramified primes: | \(23339\) | 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: | \(\Q(\sqrt{-23339}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{1649}a^{9}-\frac{406}{1649}a^{8}-\frac{64}{1649}a^{7}-\frac{401}{1649}a^{6}-\frac{446}{1649}a^{5}-\frac{318}{1649}a^{4}+\frac{465}{1649}a^{3}-\frac{814}{1649}a^{2}+\frac{683}{1649}a-\frac{264}{1649}$
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: | $a^{9}-a^{6}-a^{5}-4a^{4}-21a^{3}-10a^{2}-a+2$, $\frac{25}{1649}a^{9}-\frac{256}{1649}a^{8}+\frac{49}{1649}a^{7}-\frac{131}{1649}a^{6}+\frac{393}{1649}a^{5}+\frac{295}{1649}a^{4}+\frac{82}{1649}a^{3}+\frac{4385}{1649}a^{2}+\frac{2234}{1649}a+\frac{1645}{1649}$, $\frac{1748}{1649}a^{9}-\frac{618}{1649}a^{8}+\frac{260}{1649}a^{7}-\frac{1772}{1649}a^{6}-\frac{1280}{1649}a^{5}-\frac{6747}{1649}a^{4}-\frac{34766}{1649}a^{3}-\frac{4732}{1649}a^{2}+\frac{8}{1649}a+\frac{3546}{1649}$, $\frac{1401}{1649}a^{9}+\frac{99}{1649}a^{8}-\frac{618}{1649}a^{7}-\frac{1141}{1649}a^{6}-\frac{1524}{1649}a^{5}-\frac{5235}{1649}a^{4}-\frac{29572}{1649}a^{3}-\frac{14147}{1649}a^{2}+\frac{8708}{1649}a+\frac{2810}{1649}$, $\frac{157}{1649}a^{9}+\frac{569}{1649}a^{8}-\frac{154}{1649}a^{7}-\frac{295}{1649}a^{6}-\frac{764}{1649}a^{5}-\frac{456}{1649}a^{4}-\frac{4498}{1649}a^{3}-\frac{12368}{1649}a^{2}-\frac{4901}{1649}a-\frac{223}{1649}$, $\frac{48}{97}a^{9}-\frac{88}{97}a^{8}+\frac{32}{97}a^{7}-\frac{42}{97}a^{6}+\frac{29}{97}a^{5}-\frac{132}{97}a^{4}-\frac{669}{97}a^{3}+\frac{1183}{97}a^{2}+\frac{192}{97}a-\frac{256}{97}$ | 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: | \( 866.633051203 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 866.633051203 \cdot 1}{2\cdot\sqrt{12712961507219}}\cr\approx \mathstrut & 0.482326889119 \end{aligned}\]
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: | 5.3.23339.1 |
Degree 6 sibling: | 6.0.12712961507219.2 |
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: | 5.3.23339.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.3.0.1}{3} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(23339\) | $\Q_{23339}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $6$ | $2$ | $3$ | $3$ |