Normalized defining polynomial
\( x^{10} - 3x^{9} + 8x^{8} - 9x^{7} + 4x^{6} + 2x^{5} + 8x^{4} - 36x^{3} + 10x^{2} + 25x - 65 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[2, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(9493331265625\) \(\medspace = 5^{6}\cdot 157^{4}\) | 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: | \(19.85\) | 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: | $5^{2/3}157^{1/2}\approx 36.6378372470484$ | ||
Ramified primes: | \(5\), \(157\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{714523}a^{9}-\frac{247422}{714523}a^{8}+\frac{145801}{714523}a^{7}+\frac{185073}{714523}a^{6}+\frac{344395}{714523}a^{5}-\frac{140661}{714523}a^{4}-\frac{67794}{714523}a^{3}+\frac{96225}{714523}a^{2}+\frac{13095}{714523}a-\frac{304498}{714523}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | 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: | $\frac{95988}{714523}a^{9}-\frac{227462}{714523}a^{8}+\frac{498910}{714523}a^{7}-\frac{398225}{714523}a^{6}-\frac{333858}{714523}a^{5}-\frac{141460}{714523}a^{4}+\frac{1179535}{714523}a^{3}-\frac{2337090}{714523}a^{2}-\frac{1312143}{714523}a+\frac{3696429}{714523}$, $\frac{58639}{714523}a^{9}-\frac{189143}{714523}a^{8}+\frac{357144}{714523}a^{7}-\frac{394200}{714523}a^{6}-\frac{299667}{714523}a^{5}+\frac{233133}{714523}a^{4}+\frac{948129}{714523}a^{3}-\frac{1479402}{714523}a^{2}-\frac{234520}{714523}a+\frac{4044163}{714523}$, $\frac{76639}{714523}a^{9}-\frac{163284}{714523}a^{8}+\frac{332165}{714523}a^{7}-\frac{186426}{714523}a^{6}-\frac{391215}{714523}a^{5}-\frac{109878}{714523}a^{4}+\frac{1061413}{714523}a^{3}-\frac{1433154}{714523}a^{2}-\frac{1746156}{714523}a+\frac{1328004}{714523}$, $\frac{1863}{714523}a^{9}-\frac{79851}{714523}a^{8}+\frac{108523}{714523}a^{7}-\frac{323610}{714523}a^{6}-\frac{33769}{714523}a^{5}+\frac{178498}{714523}a^{4}+\frac{170349}{714523}a^{3}-\frac{792621}{714523}a^{2}+\frac{1531249}{714523}a+\frac{2195057}{714523}$, $\frac{59075}{714523}a^{9}-\frac{172162}{714523}a^{8}+\frac{333833}{714523}a^{7}-\frac{443471}{714523}a^{6}-\frac{193277}{714523}a^{5}-\frac{360608}{714523}a^{4}+\frac{685388}{714523}a^{3}-\frac{1682159}{714523}a^{2}-\frac{955807}{714523}a+\frac{2755267}{714523}$ | 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: | \( 454.573868293 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 454.573868293 \cdot 1}{2\cdot\sqrt{9493331265625}}\cr\approx \mathstrut & 0.459880100348 \end{aligned}\]
Galois group
A non-solvable group of order 60 |
The 5 conjugacy class representatives for $A_{5}$ |
Character table for $A_{5}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 5 sibling: | 5.1.616225.1 |
Degree 6 sibling: | 6.2.15405625.2 |
Degree 12 sibling: | data not computed |
Degree 15 sibling: | data not computed |
Degree 20 sibling: | data not computed |
Degree 30 sibling: | data not computed |
Minimal sibling: | 5.1.616225.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/7.3.0.1}{3} }^{3}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{4}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.3.0.1}{3} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.3.0.1}{3} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.3.0.1}{3} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
5.3.2.1 | $x^{3} + 5$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
5.6.4.1 | $x^{6} + 12 x^{5} + 54 x^{4} + 122 x^{3} + 168 x^{2} + 228 x + 233$ | $3$ | $2$ | $4$ | $S_3$ | $[\ ]_{3}^{2}$ | |
\(157\) | 157.2.1.2 | $x^{2} + 314$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
157.2.1.1 | $x^{2} + 157$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
157.2.0.1 | $x^{2} + 152 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
157.4.2.1 | $x^{4} + 32018 x^{3} + 258698669 x^{2} + 38591103292 x + 1651335445$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |