Normalized defining polynomial
\( x^{10} + 858x^{8} + 267696x^{6} + 36540504x^{4} + 2035828080x^{2} + 31758918048 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[0, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-6971193481768943910912\) \(\medspace = -\,2^{15}\cdot 3^{5}\cdot 11^{9}\cdot 13^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(152.87\) | 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: | $2^{3/2}3^{1/2}11^{9/10}13^{1/2}\approx 152.87297371889034$ | ||
Ramified primes: | \(2\), \(3\), \(11\), \(13\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-858}) \) | ||
$\card{ \Gal(K/\Q) }$: | $10$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(3432=2^{3}\cdot 3\cdot 11\cdot 13\) | ||
Dirichlet character group: | $\lbrace$$\chi_{3432}(1,·)$, $\chi_{3432}(2339,·)$, $\chi_{3432}(1873,·)$, $\chi_{3432}(3275,·)$, $\chi_{3432}(625,·)$, $\chi_{3432}(1091,·)$, $\chi_{3432}(1715,·)$, $\chi_{3432}(2809,·)$, $\chi_{3432}(313,·)$, $\chi_{3432}(1403,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | \(\Q(\sqrt{-858}) \), 10.0.6971193481768943910912.1$^{15}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{78}a^{2}$, $\frac{1}{78}a^{3}$, $\frac{1}{6084}a^{4}$, $\frac{1}{6084}a^{5}$, $\frac{1}{474552}a^{6}$, $\frac{1}{474552}a^{7}$, $\frac{1}{37015056}a^{8}$, $\frac{1}{37015056}a^{9}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{2}\times C_{40804}$, which has order $163216$ (assuming GRH)
Unit group
Rank: | $4$ | 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: | $\frac{1}{37015056}a^{8}+\frac{1}{59319}a^{6}+\frac{7}{2028}a^{4}+\frac{10}{39}a^{2}+5$, $\frac{1}{474552}a^{6}+\frac{1}{1014}a^{4}+\frac{3}{26}a^{2}+2$, $\frac{1}{474552}a^{6}+\frac{1}{1014}a^{4}+\frac{3}{26}a^{2}+3$, $\frac{1}{78}a^{2}+2$ (assuming GRH) | 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: | \( 26.1711060094 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 26.1711060094 \cdot 163216}{2\cdot\sqrt{6971193481768943910912}}\cr\approx \mathstrut & 0.250495878530 \end{aligned}\] (assuming GRH)
Galois group
A cyclic group of order 10 |
The 10 conjugacy class representatives for $C_{10}$ |
Character table for $C_{10}$ |
Intermediate fields
\(\Q(\sqrt{-858}) \), \(\Q(\zeta_{11})^+\) |
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 | R | R | ${\href{/padicField/5.10.0.1}{10} }$ | ${\href{/padicField/7.10.0.1}{10} }$ | R | R | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.1.0.1}{1} }^{10}$ | ${\href{/padicField/29.10.0.1}{10} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }$ | ${\href{/padicField/43.2.0.1}{2} }^{5}$ | ${\href{/padicField/47.10.0.1}{10} }$ | ${\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.10.15.13 | $x^{10} - 12 x^{9} + 174 x^{8} - 640 x^{7} - 40 x^{6} + 11424 x^{5} - 7984 x^{4} - 79360 x^{3} + 84112 x^{2} + 955968 x + 1404384$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ |
\(3\) | 3.10.5.2 | $x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
\(11\) | 11.10.9.7 | $x^{10} + 11$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
\(13\) | 13.10.5.1 | $x^{10} + 650 x^{9} + 169065 x^{8} + 22003800 x^{7} + 1434642698 x^{6} + 37701182242 x^{5} + 18651037600 x^{4} + 3808243140 x^{3} + 6315953361 x^{2} + 164195122608 x + 421659070668$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |