Normalized defining polynomial
\( x^{10} - 5x^{8} - 10x^{7} + 10x^{6} + 42x^{5} - 5x^{4} - 140x^{3} - 240x^{2} - 200x - 68 \)
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: | \(625000000000000\) \(\medspace = 2^{12}\cdot 5^{16}\) | 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: | \(30.17\) | 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: | $2^{3/2}5^{8/5}\approx 37.14471242937835$ | ||
Ramified primes: | \(2\), \(5\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{6}a^{8}-\frac{1}{6}a^{6}+\frac{1}{3}a^{4}-\frac{1}{2}a^{2}-\frac{1}{3}$, $\frac{1}{43572}a^{9}+\frac{628}{10893}a^{8}-\frac{539}{43572}a^{7}-\frac{541}{7262}a^{6}-\frac{3310}{10893}a^{5}-\frac{6701}{21786}a^{4}-\frac{13721}{43572}a^{3}-\frac{460}{10893}a^{2}+\frac{9049}{21786}a+\frac{1365}{3631}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{5}$, which has order $5$
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: | $a+1$, $\frac{1659}{14524}a^{9}-\frac{245}{3631}a^{8}-\frac{8237}{14524}a^{7}-\frac{5617}{7262}a^{6}+\frac{6044}{3631}a^{5}+\frac{30211}{7262}a^{4}-\frac{47603}{14524}a^{3}-\frac{51464}{3631}a^{2}-\frac{136241}{7262}a-\frac{36306}{3631}$, $\frac{471}{14524}a^{9}-\frac{277}{7262}a^{8}-\frac{6359}{43572}a^{7}-\frac{1069}{10893}a^{6}+\frac{2320}{3631}a^{5}+\frac{2799}{7262}a^{4}-\frac{56329}{43572}a^{3}-\frac{61789}{21786}a^{2}-\frac{74765}{21786}a-\frac{16103}{10893}$, $\frac{10187}{43572}a^{9}-\frac{4403}{21786}a^{8}-\frac{44293}{43572}a^{7}-\frac{5101}{3631}a^{6}+\frac{38437}{10893}a^{5}+\frac{144953}{21786}a^{4}-\frac{301855}{43572}a^{3}-\frac{581389}{21786}a^{2}-\frac{713341}{21786}a-\frac{66833}{3631}$, $\frac{9575}{10893}a^{9}-\frac{16843}{21786}a^{8}-\frac{41215}{10893}a^{7}-\frac{118003}{21786}a^{6}+\frac{152236}{10893}a^{5}+\frac{271346}{10893}a^{4}-\frac{303106}{10893}a^{3}-\frac{2197531}{21786}a^{2}-\frac{1282868}{10893}a-\frac{667018}{10893}$ | 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: | \( 2089.87951472 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 2089.87951472 \cdot 5}{2\cdot\sqrt{625000000000000}}\cr\approx \mathstrut & 1.30286888896 \end{aligned}\]
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: | 5.1.25000000.2 |
Degree 6 sibling: | 6.2.25000000.3 |
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: | 5.1.25000000.2 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{3}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | R | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.3.0.1}{3} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.3.0.1}{3} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.4.6.7 | $x^{4} + 2 x^{3} + 2 x^{2} + 2$ | $4$ | $1$ | $6$ | $A_4$ | $[2, 2]^{3}$ |
2.6.6.1 | $x^{6} + 6 x^{5} + 34 x^{4} + 80 x^{3} + 204 x^{2} + 216 x + 216$ | $2$ | $3$ | $6$ | $A_4$ | $[2, 2]^{3}$ | |
\(5\) | 5.5.8.4 | $x^{5} + 20 x^{4} + 55$ | $5$ | $1$ | $8$ | $C_5$ | $[2]$ |
5.5.8.4 | $x^{5} + 20 x^{4} + 55$ | $5$ | $1$ | $8$ | $C_5$ | $[2]$ |