Normalized defining polynomial
\( x^{10} - 30x^{8} - 10x^{7} + 240x^{6} + 78x^{5} - 655x^{4} - 320x^{3} + 570x^{2} + 420x + 49 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[10, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(5000000000000000\) \(\medspace = 2^{15}\cdot 5^{16}\) | 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: | \(37.14\) | 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: | \(\Q(\sqrt{2}) \) | ||
$\card{ \Gal(K/\Q) }$: | $10$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(200=2^{3}\cdot 5^{2}\) | ||
Dirichlet character group: | $\lbrace$$\chi_{200}(1,·)$, $\chi_{200}(101,·)$, $\chi_{200}(161,·)$, $\chi_{200}(41,·)$, $\chi_{200}(141,·)$, $\chi_{200}(81,·)$, $\chi_{200}(21,·)$, $\chi_{200}(121,·)$, $\chi_{200}(61,·)$, $\chi_{200}(181,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7}a^{6}-\frac{3}{7}a^{5}+\frac{2}{7}a^{4}+\frac{1}{7}a^{3}-\frac{3}{7}a^{2}+\frac{2}{7}a$, $\frac{1}{7}a^{7}-\frac{1}{7}a$, $\frac{1}{7}a^{8}-\frac{1}{7}a^{2}$, $\frac{1}{359807}a^{9}-\frac{839}{359807}a^{8}-\frac{15723}{359807}a^{7}-\frac{18470}{359807}a^{6}-\frac{26532}{359807}a^{5}+\frac{106595}{359807}a^{4}+\frac{55281}{359807}a^{3}-\frac{17377}{359807}a^{2}-\frac{2573}{51401}a+\frac{3142}{7343}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $7$ |
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)
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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: | $\frac{17490}{359807}a^{9}-\frac{24825}{359807}a^{8}-\frac{513930}{359807}a^{7}+\frac{528995}{359807}a^{6}+\frac{4115628}{359807}a^{5}-\frac{3565790}{359807}a^{4}-\frac{10268320}{359807}a^{3}+\frac{4225265}{359807}a^{2}+\frac{1295845}{51401}a+\frac{27940}{7343}$, $\frac{3380}{51401}a^{9}-\frac{1422}{51401}a^{8}-\frac{97908}{51401}a^{7}+\frac{8929}{51401}a^{6}+\frac{728856}{51401}a^{5}-\frac{110683}{51401}a^{4}-\frac{1703774}{51401}a^{3}-\frac{49004}{51401}a^{2}+\frac{184094}{7343}a+\frac{7227}{1049}$, $\frac{24340}{359807}a^{9}-\frac{15063}{359807}a^{8}-\frac{685588}{359807}a^{7}+\frac{198950}{359807}a^{6}+\frac{4742176}{359807}a^{5}-\frac{1845012}{359807}a^{4}-\frac{8774008}{359807}a^{3}+\frac{1718882}{359807}a^{2}+\frac{611096}{51401}a+\frac{6278}{7343}$, $\frac{17490}{359807}a^{9}-\frac{24825}{359807}a^{8}-\frac{513930}{359807}a^{7}+\frac{528995}{359807}a^{6}+\frac{4115628}{359807}a^{5}-\frac{3565790}{359807}a^{4}-\frac{10268320}{359807}a^{3}+\frac{4225265}{359807}a^{2}+\frac{1244444}{51401}a+\frac{35283}{7343}$, $\frac{680}{359807}a^{9}-\frac{5109}{359807}a^{8}-\frac{232}{359807}a^{7}+\frac{136447}{359807}a^{6}-\frac{359816}{359807}a^{5}-\frac{1070231}{359807}a^{4}+\frac{3152410}{359807}a^{3}+\frac{2061910}{359807}a^{2}-\frac{677562}{51401}a-\frac{58997}{7343}$, $\frac{2589}{51401}a^{9}+\frac{8700}{51401}a^{8}-\frac{63342}{51401}a^{7}-\frac{250876}{51401}a^{6}+\frac{171306}{51401}a^{5}+\frac{1228767}{51401}a^{4}+\frac{352560}{51401}a^{3}-\frac{1540522}{51401}a^{2}-\frac{154550}{7343}a-\frac{2455}{1049}$, $\frac{33452}{359807}a^{9}-\frac{52683}{359807}a^{8}-\frac{955982}{359807}a^{7}+\frac{1163803}{359807}a^{6}+\frac{7188743}{359807}a^{5}-\frac{8198392}{359807}a^{4}-\frac{15465466}{359807}a^{3}+\frac{11255324}{359807}a^{2}+\frac{1603424}{51401}a+\frac{27854}{7343}$, $\frac{95336}{359807}a^{9}+\frac{95854}{359807}a^{8}-\frac{2582016}{359807}a^{7}-\frac{3455730}{359807}a^{6}+\frac{14432139}{359807}a^{5}+\frac{17788159}{359807}a^{4}-\frac{14832805}{359807}a^{3}-\frac{25802744}{359807}a^{2}-\frac{1467870}{51401}a-\frac{26659}{7343}$, $\frac{6936}{359807}a^{9}-\frac{10991}{359807}a^{8}-\frac{187410}{359807}a^{7}+\frac{240377}{359807}a^{6}+\frac{1223252}{359807}a^{5}-\frac{1705297}{359807}a^{4}-\frac{2027083}{359807}a^{3}+\frac{1344899}{359807}a^{2}+\frac{290882}{51401}a+\frac{13574}{7343}$ | 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: | \( 82605.472142 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{0}\cdot 82605.472142 \cdot 1}{2\cdot\sqrt{5000000000000000}}\cr\approx \mathstrut & 0.59812750863 \end{aligned}\]
Galois group
A cyclic group of order 10 |
The 10 conjugacy class representatives for $C_{10}$ |
Character table for $C_{10}$ |
Intermediate fields
\(\Q(\sqrt{2}) \), 5.5.390625.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.10.0.1}{10} }$ | R | ${\href{/padicField/7.1.0.1}{1} }^{10}$ | ${\href{/padicField/11.10.0.1}{10} }$ | ${\href{/padicField/13.10.0.1}{10} }$ | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.10.0.1}{10} }$ | ${\href{/padicField/23.5.0.1}{5} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.10.0.1}{10} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{5}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }$ | ${\href{/padicField/59.10.0.1}{10} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.10.15.1 | $x^{10} + 70 x^{8} + 2 x^{7} + 1960 x^{6} - 26 x^{5} + 27441 x^{4} - 2240 x^{3} + 192110 x^{2} - 14504 x + 537993$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ |
\(5\) | 5.10.16.7 | $x^{10} + 40 x^{9} + 400 x^{8} + 10 x^{5} + 200 x^{4} - 1225$ | $5$ | $2$ | $16$ | $C_{10}$ | $[2]^{2}$ |