Normalized defining polynomial
\( x^{10} - x^{9} + x^{8} + 3x^{7} - 4x^{6} - x^{5} - 15x^{4} + 10x^{3} - x^{2} - x + 1 \)
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: | \(-1374665998159\) \(\medspace = -\,11119^{3}\) | 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: | \(16.36\) | 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: | $11119^{1/2}\approx 105.4466689848475$ | ||
Ramified primes: | \(11119\) | 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: | \(\Q(\sqrt{-11119}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{4331}a^{9}+\frac{2015}{4331}a^{8}-\frac{237}{4331}a^{7}-\frac{1379}{4331}a^{6}+\frac{434}{4331}a^{5}+\frac{81}{4331}a^{4}-\frac{1297}{4331}a^{3}+\frac{1182}{4331}a^{2}+\frac{861}{4331}a-\frac{956}{4331}$
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: | $\frac{4117}{4331}a^{9}-\frac{2441}{4331}a^{8}+\frac{3077}{4331}a^{7}+\frac{13591}{4331}a^{6}-\frac{10587}{4331}a^{5}-\frac{8672}{4331}a^{4}-\frac{64591}{4331}a^{3}+\frac{15574}{4331}a^{2}+\frac{1979}{4331}a-\frac{3304}{4331}$, $a^{9}-a^{8}+a^{7}+3a^{6}-4a^{5}-a^{4}-15a^{3}+10a^{2}-a-1$, $\frac{163}{4331}a^{9}-\frac{711}{4331}a^{8}+\frac{348}{4331}a^{7}+\frac{435}{4331}a^{6}-\frac{2885}{4331}a^{5}+\frac{210}{4331}a^{4}+\frac{808}{4331}a^{3}+\frac{10764}{4331}a^{2}-\frac{2580}{4331}a+\frac{88}{4331}$, $\frac{1754}{4331}a^{9}+\frac{214}{4331}a^{8}+\frac{78}{4331}a^{7}+\frac{6594}{4331}a^{6}-\frac{1020}{4331}a^{5}-\frac{9511}{4331}a^{4}-\frac{31480}{4331}a^{3}-\frac{14314}{4331}a^{2}+\frac{11668}{4331}a+\frac{3604}{4331}$, $\frac{2655}{4331}a^{9}-\frac{3291}{4331}a^{8}+\frac{3091}{4331}a^{7}+\frac{7112}{4331}a^{6}-\frac{12769}{4331}a^{5}-\frac{1495}{4331}a^{4}-\frac{39369}{4331}a^{3}+\frac{37214}{4331}a^{2}-\frac{5144}{4331}a-\frac{214}{4331}$, $\frac{687}{4331}a^{9}-\frac{1615}{4331}a^{8}+\frac{1759}{4331}a^{7}+\frac{1116}{4331}a^{6}-\frac{5012}{4331}a^{5}+\frac{3675}{4331}a^{4}-\frac{7515}{4331}a^{3}+\frac{23792}{4331}a^{2}-\frac{10502}{4331}a+\frac{5871}{4331}$ | 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: | \( 232.004719978 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 232.004719978 \cdot 1}{2\cdot\sqrt{1374665998159}}\cr\approx \mathstrut & 0.392670085624 \end{aligned}\]
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: | 5.3.11119.1 |
Degree 6 sibling: | 6.0.1374665998159.2 |
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: | 5.3.11119.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{3}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.3.0.1}{3} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(11119\) | $\Q_{11119}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $6$ | $2$ | $3$ | $3$ |