Normalized defining polynomial
\( x^{10} - 5x^{9} - 3x^{8} + 31x^{7} + 29x^{6} - 172x^{5} + 151x^{4} - 45x^{3} + 117x^{2} - 91x - 24 \)
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: | \(-9903331494610723\) \(\medspace = -\,13^{3}\cdot 16519^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(39.77\) | 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: | $13^{1/2}16519^{1/2}\approx 463.4080275523936$ | ||
Ramified primes: | \(13\), \(16519\) | 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: | \(\Q(\sqrt{-214747}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{19}a^{7}+\frac{3}{19}a^{6}-\frac{3}{19}a^{4}-\frac{3}{19}a^{3}-\frac{6}{19}a-\frac{9}{19}$, $\frac{1}{19}a^{8}-\frac{9}{19}a^{6}-\frac{3}{19}a^{5}+\frac{6}{19}a^{4}+\frac{9}{19}a^{3}-\frac{6}{19}a^{2}+\frac{9}{19}a+\frac{8}{19}$, $\frac{1}{250211}a^{9}-\frac{2549}{250211}a^{8}+\frac{5505}{250211}a^{7}-\frac{19211}{250211}a^{6}+\frac{15823}{250211}a^{5}-\frac{48927}{250211}a^{4}+\frac{102403}{250211}a^{3}+\frac{22219}{250211}a^{2}+\frac{114850}{250211}a-\frac{37395}{250211}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}$, which has order $2$
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: | $\frac{3272}{250211}a^{9}-\frac{17520}{250211}a^{8}-\frac{2832}{250211}a^{7}+\frac{102597}{250211}a^{6}+\frac{31855}{250211}a^{5}-\frac{559878}{250211}a^{4}+\frac{872903}{250211}a^{3}-\frac{505903}{250211}a^{2}+\frac{64461}{250211}a+\frac{23077}{250211}$, $\frac{11387}{250211}a^{9}-\frac{53663}{250211}a^{8}-\frac{51681}{250211}a^{7}+\frac{350165}{250211}a^{6}+\frac{22780}{13169}a^{5}-\frac{1926709}{250211}a^{4}+\frac{1159559}{250211}a^{3}+\frac{360488}{250211}a^{2}+\frac{826376}{250211}a-\frac{1221967}{250211}$, $\frac{478}{250211}a^{9}-\frac{6874}{250211}a^{8}+\frac{10759}{250211}a^{7}+\frac{74949}{250211}a^{6}-\frac{74626}{250211}a^{5}-\frac{499384}{250211}a^{4}+\frac{407700}{250211}a^{3}+\frac{849284}{250211}a^{2}-\frac{793401}{250211}a-\frac{109829}{250211}$, $\frac{7278}{250211}a^{9}-\frac{22839}{250211}a^{8}-\frac{73722}{250211}a^{7}+\frac{116136}{250211}a^{6}+\frac{523649}{250211}a^{5}-\frac{396016}{250211}a^{4}-\frac{405591}{250211}a^{3}-\frac{255649}{250211}a^{2}+\frac{173560}{250211}a+\frac{371645}{250211}$, $\frac{18775}{250211}a^{9}-\frac{67174}{250211}a^{8}-\frac{151965}{250211}a^{7}+\frac{354779}{250211}a^{6}+\frac{1077212}{250211}a^{5}-\frac{1567941}{250211}a^{4}+\frac{26768}{13169}a^{3}-\frac{940856}{250211}a^{2}+\frac{1017534}{250211}a+\frac{290659}{250211}$, $\frac{18315}{250211}a^{9}-\frac{66675}{250211}a^{8}-\frac{142648}{250211}a^{7}+\frac{342271}{250211}a^{6}+\frac{1067920}{250211}a^{5}-\frac{1475159}{250211}a^{4}+\frac{284852}{250211}a^{3}-\frac{2127662}{250211}a^{2}+\frac{1955561}{250211}a+\frac{504349}{250211}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 25771.8295197 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 25771.8295197 \cdot 2}{2\cdot\sqrt{9903331494610723}}\cr\approx \mathstrut & 1.02781317253 \end{aligned}\]
Galois group
A non-solvable group of order 720 |
The 11 conjugacy class representatives for $S_{6}$ |
Character table for $S_{6}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 6 siblings: | 6.4.214747.1, 6.0.9903331494610723.1 |
Degree 12 siblings: | data not computed |
Degree 15 siblings: | data not computed |
Degree 20 siblings: | data not computed |
Degree 30 siblings: | data not computed |
Degree 36 sibling: | data not computed |
Degree 40 siblings: | data not computed |
Degree 45 sibling: | data not computed |
Minimal sibling: | 6.4.214747.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.3.0.1}{3} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | R | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.3.0.1}{3} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(13\) | 13.2.1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
13.4.0.1 | $x^{4} + 3 x^{2} + 12 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
13.4.2.2 | $x^{4} - 156 x^{2} + 338$ | $2$ | $2$ | $2$ | $C_4$ | $[\ ]_{2}^{2}$ | |
\(16519\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $2$ | $2$ | $2$ |