Normalized defining polynomial
\( x^{10} - 4x^{9} - 12x^{8} + 53x^{7} + 22x^{6} - 151x^{5} - 11x^{4} + 144x^{3} + 11x^{2} - 41x - 7 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[10, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(8643523020765625\) \(\medspace = 5^{6}\cdot 8209^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(39.23\) | 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: | $5^{2/3}8209^{1/2}\approx 264.92633453725404$ | ||
Ramified primes: | \(5\), \(8209\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{8209}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{48821}a^{9}+\frac{12960}{48821}a^{8}+\frac{20367}{48821}a^{7}+\frac{13873}{48821}a^{6}-\frac{6970}{48821}a^{5}+\frac{8440}{48821}a^{4}+\frac{8288}{48821}a^{3}-\frac{9245}{48821}a^{2}+\frac{3386}{48821}a+\frac{5984}{48821}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
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)
| |
Fundamental units: | $\frac{2892}{48821}a^{9}-\frac{14208}{48821}a^{8}-\frac{25583}{48821}a^{7}+\frac{185138}{48821}a^{6}-\frac{42988}{48821}a^{5}-\frac{490230}{48821}a^{4}+\frac{144248}{48821}a^{3}+\frac{359115}{48821}a^{2}+\frac{76933}{48821}a+\frac{23094}{48821}$, $\frac{2892}{48821}a^{9}-\frac{14208}{48821}a^{8}-\frac{25583}{48821}a^{7}+\frac{185138}{48821}a^{6}-\frac{42988}{48821}a^{5}-\frac{490230}{48821}a^{4}+\frac{144248}{48821}a^{3}+\frac{407936}{48821}a^{2}-\frac{20709}{48821}a-\frac{123369}{48821}$, $\frac{8172}{48821}a^{9}-\frac{32450}{48821}a^{8}-\frac{89307}{48821}a^{7}+\frac{398362}{48821}a^{6}+\frac{64088}{48821}a^{5}-\frac{842350}{48821}a^{4}+\frac{210093}{48821}a^{3}+\frac{415336}{48821}a^{2}-\frac{108757}{48821}a-\frac{17394}{48821}$, $\frac{2892}{48821}a^{9}-\frac{14208}{48821}a^{8}-\frac{25583}{48821}a^{7}+\frac{185138}{48821}a^{6}-\frac{42988}{48821}a^{5}-\frac{490230}{48821}a^{4}+\frac{144248}{48821}a^{3}+\frac{359115}{48821}a^{2}-\frac{20709}{48821}a-\frac{25727}{48821}$, $\frac{8424}{48821}a^{9}-\frac{37537}{48821}a^{8}-\frac{83028}{48821}a^{7}+\frac{476888}{48821}a^{6}-\frac{32438}{48821}a^{5}-\frac{1156520}{48821}a^{4}+\frac{345829}{48821}a^{3}+\frac{770751}{48821}a^{2}-\frac{231905}{48821}a-\frac{71698}{48821}$, $\frac{14536}{48821}a^{9}-\frac{62500}{48821}a^{8}-\frac{142295}{48821}a^{7}+\frac{759513}{48821}a^{6}-\frac{61166}{48821}a^{5}-\frac{1516784}{48821}a^{4}+\frac{521171}{48821}a^{3}+\frac{555924}{48821}a^{2}-\frac{139135}{48821}a-\frac{15598}{48821}$, $\frac{5532}{48821}a^{9}-\frac{23329}{48821}a^{8}-\frac{57445}{48821}a^{7}+\frac{291750}{48821}a^{6}+\frac{10550}{48821}a^{5}-\frac{666290}{48821}a^{4}+\frac{201581}{48821}a^{3}+\frac{362815}{48821}a^{2}-\frac{162375}{48821}a+\frac{2850}{48821}$, $\frac{2761}{48821}a^{9}-\frac{3233}{48821}a^{8}-\frac{57326}{48821}a^{7}+\frac{27689}{48821}a^{6}+\frac{381872}{48821}a^{5}+\frac{64044}{48821}a^{4}-\frac{941480}{48821}a^{3}-\frac{382630}{48821}a^{2}+\frac{707429}{48821}a+\frac{362073}{48821}$, $\frac{121667}{48821}a^{9}-\frac{602190}{48821}a^{8}-\frac{894486}{48821}a^{7}+\frac{7321008}{48821}a^{6}-\frac{4196826}{48821}a^{5}-\frac{14677734}{48821}a^{4}+\frac{12427696}{48821}a^{3}+\frac{6371155}{48821}a^{2}-\frac{4576310}{48821}a-\frac{939844}{48821}$ | 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: | \( 99378.8210563 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{0}\cdot 99378.8210563 \cdot 1}{2\cdot\sqrt{8643523020765625}}\cr\approx \mathstrut & 0.547291124040 \end{aligned}\]
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: | 5.5.205225.1 |
Degree 6 sibling: | 6.6.345740920830625.1 |
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: | 5.5.205225.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | ${\href{/padicField/3.3.0.1}{3} }^{3}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | R | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.3.0.1}{3} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
5.3.2.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(8209\) | $\Q_{8209}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $6$ | $2$ | $3$ | $3$ |