Normalized defining polynomial
\( x^{10} - 3 x^{9} + 44 x^{8} - 100 x^{7} + 910 x^{6} - 1500 x^{5} + 10534 x^{4} - 11333 x^{3} + \cdots + 200509 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-19340358336761319\) \(\medspace = -\,3^{5}\cdot 11^{8}\cdot 13^{5}\) | 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: | \(42.53\) | 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: | $3^{1/2}11^{4/5}13^{1/2}\approx 42.52520850150444$ | ||
Ramified primes: | \(3\), \(11\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-39}) \) | ||
$\card{ \Gal(K/\Q) }$: | $10$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(429=3\cdot 11\cdot 13\) | ||
Dirichlet character group: | $\lbrace$$\chi_{429}(1,·)$, $\chi_{429}(196,·)$, $\chi_{429}(389,·)$, $\chi_{429}(38,·)$, $\chi_{429}(235,·)$, $\chi_{429}(311,·)$, $\chi_{429}(313,·)$, $\chi_{429}(155,·)$, $\chi_{429}(157,·)$, $\chi_{429}(350,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | \(\Q(\sqrt{-39}) \), 10.0.19340358336761319.3$^{15}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{22\!\cdots\!43}a^{9}+\frac{891427302035879}{22\!\cdots\!43}a^{8}+\frac{88\!\cdots\!73}{22\!\cdots\!43}a^{7}-\frac{18\!\cdots\!06}{22\!\cdots\!43}a^{6}-\frac{77\!\cdots\!35}{22\!\cdots\!43}a^{5}-\frac{902282174506390}{22\!\cdots\!43}a^{4}+\frac{10\!\cdots\!60}{22\!\cdots\!43}a^{3}+\frac{10\!\cdots\!94}{22\!\cdots\!43}a^{2}+\frac{793623512519196}{22\!\cdots\!43}a+\frac{55\!\cdots\!77}{22\!\cdots\!43}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{220}$, which has order $220$
Unit group
Rank: | $4$ | 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)
| |
Fundamental units: | $\frac{1966882096530}{22\!\cdots\!43}a^{9}-\frac{17829181392672}{22\!\cdots\!43}a^{8}+\frac{109595581278124}{22\!\cdots\!43}a^{7}-\frac{553899195829091}{22\!\cdots\!43}a^{6}+\frac{20\!\cdots\!35}{22\!\cdots\!43}a^{5}-\frac{80\!\cdots\!29}{22\!\cdots\!43}a^{4}+\frac{18\!\cdots\!09}{22\!\cdots\!43}a^{3}-\frac{59\!\cdots\!71}{22\!\cdots\!43}a^{2}+\frac{69\!\cdots\!19}{22\!\cdots\!43}a-\frac{18\!\cdots\!38}{22\!\cdots\!43}$, $\frac{160221135214}{22\!\cdots\!43}a^{9}-\frac{1867293229626}{22\!\cdots\!43}a^{8}+\frac{10682091604080}{22\!\cdots\!43}a^{7}-\frac{85960745566395}{22\!\cdots\!43}a^{6}+\frac{282571846206003}{22\!\cdots\!43}a^{5}-\frac{20\!\cdots\!74}{22\!\cdots\!43}a^{4}+\frac{41\!\cdots\!19}{22\!\cdots\!43}a^{3}-\frac{22\!\cdots\!40}{22\!\cdots\!43}a^{2}+\frac{22\!\cdots\!13}{22\!\cdots\!43}a-\frac{12\!\cdots\!42}{22\!\cdots\!43}$, $\frac{2219290768256}{22\!\cdots\!43}a^{9}-\frac{5140790154273}{22\!\cdots\!43}a^{8}+\frac{86509589322526}{22\!\cdots\!43}a^{7}-\frac{135356391774990}{22\!\cdots\!43}a^{6}+\frac{13\!\cdots\!42}{22\!\cdots\!43}a^{5}-\frac{843274343217020}{22\!\cdots\!43}a^{4}+\frac{10\!\cdots\!82}{22\!\cdots\!43}a^{3}+\frac{37\!\cdots\!88}{22\!\cdots\!43}a^{2}+\frac{31\!\cdots\!77}{22\!\cdots\!43}a+\frac{51\!\cdots\!90}{22\!\cdots\!43}$, $\frac{215898129096}{22\!\cdots\!43}a^{9}+\frac{846884501244}{22\!\cdots\!43}a^{8}+\frac{6290598276182}{22\!\cdots\!43}a^{7}+\frac{51494118273486}{22\!\cdots\!43}a^{6}+\frac{85443743452840}{22\!\cdots\!43}a^{5}+\frac{15\!\cdots\!94}{22\!\cdots\!43}a^{4}+\frac{567971786804618}{22\!\cdots\!43}a^{3}+\frac{18\!\cdots\!19}{22\!\cdots\!43}a^{2}+\frac{15\!\cdots\!45}{22\!\cdots\!43}a+\frac{99\!\cdots\!93}{22\!\cdots\!43}$ | 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: | \( 26.1711060094 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 26.1711060094 \cdot 220}{2\cdot\sqrt{19340358336761319}}\cr\approx \mathstrut & 0.202713114098 \end{aligned}\]
Galois group
A cyclic group of order 10 |
The 10 conjugacy class representatives for $C_{10}$ |
Character table for $C_{10}$ |
Intermediate fields
\(\Q(\sqrt{-39}) \), \(\Q(\zeta_{11})^+\) |
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 | ${\href{/padicField/2.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.10.0.1}{10} }$ | R | R | ${\href{/padicField/17.10.0.1}{10} }$ | ${\href{/padicField/19.10.0.1}{10} }$ | ${\href{/padicField/23.2.0.1}{2} }^{5}$ | ${\href{/padicField/29.10.0.1}{10} }$ | ${\href{/padicField/31.10.0.1}{10} }$ | ${\href{/padicField/37.10.0.1}{10} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.1.0.1}{1} }^{10}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.10.5.2 | $x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
\(11\) | 11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ |
11.5.4.4 | $x^{5} + 11$ | $5$ | $1$ | $4$ | $C_5$ | $[\ ]_{5}$ | |
\(13\) | 13.10.5.1 | $x^{10} + 650 x^{9} + 169065 x^{8} + 22003800 x^{7} + 1434642698 x^{6} + 37701182242 x^{5} + 18651037600 x^{4} + 3808243140 x^{3} + 6315953361 x^{2} + 164195122608 x + 421659070668$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |