Normalized defining polynomial
\( x^{10} + 374x^{8} + 50864x^{6} + 3026408x^{4} + 73498480x^{2} + 499789664 \)
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: | \(-109705577585055727616\) \(\medspace = -\,2^{15}\cdot 11^{9}\cdot 17^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(100.93\) | 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: | $2^{3/2}11^{9/10}17^{1/2}\approx 100.93060367784076$ | ||
Ramified primes: | \(2\), \(11\), \(17\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-374}) \) | ||
$\card{ \Gal(K/\Q) }$: | $10$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(1496=2^{3}\cdot 11\cdot 17\) | ||
Dirichlet character group: | $\lbrace$$\chi_{1496}(1,·)$, $\chi_{1496}(101,·)$, $\chi_{1496}(273,·)$, $\chi_{1496}(137,·)$, $\chi_{1496}(237,·)$, $\chi_{1496}(817,·)$, $\chi_{1496}(373,·)$, $\chi_{1496}(1225,·)$, $\chi_{1496}(1053,·)$, $\chi_{1496}(645,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | \(\Q(\sqrt{-374}) \), 10.0.109705577585055727616.1$^{15}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{34}a^{2}$, $\frac{1}{34}a^{3}$, $\frac{1}{1156}a^{4}$, $\frac{1}{1156}a^{5}$, $\frac{1}{39304}a^{6}$, $\frac{1}{39304}a^{7}$, $\frac{1}{1336336}a^{8}$, $\frac{1}{1336336}a^{9}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{2}\times C_{7294}$, which has order $14588$ (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: | $\frac{1}{1336336}a^{8}+\frac{9}{39304}a^{6}+\frac{27}{1156}a^{4}+\frac{15}{17}a^{2}+9$, $\frac{1}{34}a^{2}+2$, $\frac{1}{39304}a^{6}+\frac{3}{578}a^{4}+\frac{5}{17}a^{2}+4$, $\frac{1}{39304}a^{6}+\frac{3}{578}a^{4}+\frac{9}{34}a^{2}+2$ (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
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 26.1711060094 \cdot 14588}{2\cdot\sqrt{109705577585055727616}}\cr\approx \mathstrut & 0.178473123027 \end{aligned}\] (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
\(\Q(\sqrt{-374}) \), \(\Q(\zeta_{11})^+\) |
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 | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{5}$ | ${\href{/padicField/29.10.0.1}{10} }$ | ${\href{/padicField/31.10.0.1}{10} }$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.1.0.1}{1} }^{10}$ | ${\href{/padicField/47.5.0.1}{5} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }$ | ${\href{/padicField/59.10.0.1}{10} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.10.15.5 | $x^{10} + 46 x^{8} + 808 x^{6} + 6768 x^{4} + 27216 x^{2} + 9568$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ |
\(11\) | 11.10.9.7 | $x^{10} + 11$ | $10$ | $1$ | $9$ | $C_{10}$ | $[\ ]_{10}$ |
\(17\) | 17.10.5.2 | $x^{10} + 83521 x^{2} - 19877998$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |