Normalized defining polynomial
\( x^{10} - 10x^{8} - 10x^{7} + 80x^{6} + 118x^{5} + 305x^{4} + 560x^{3} + 1410x^{2} + 260x + 2393 \)
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: | \(-5000000000000000\) \(\medspace = -\,2^{15}\cdot 5^{16}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
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))
| |
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)))
| |
Discriminant root field: | \(\Q(\sqrt{-2}) \) | ||
$\card{ \Gal(K/\Q) }$: | $10$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(200=2^{3}\cdot 5^{2}\) | ||
Dirichlet character group: | $\lbrace$$\chi_{200}(1,·)$, $\chi_{200}(131,·)$, $\chi_{200}(161,·)$, $\chi_{200}(41,·)$, $\chi_{200}(11,·)$, $\chi_{200}(81,·)$, $\chi_{200}(51,·)$, $\chi_{200}(171,·)$, $\chi_{200}(121,·)$, $\chi_{200}(91,·)$$\rbrace$ | ||
This is a CM field. | |||
Reflex fields: | \(\Q(\sqrt{-2}) \), 10.0.5000000000000000.1$^{15}$ |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{7}a^{8}+\frac{1}{7}a^{7}+\frac{1}{7}a^{6}+\frac{1}{7}a^{5}+\frac{2}{7}a^{3}-\frac{1}{7}a^{2}-\frac{2}{7}a-\frac{2}{7}$, $\frac{1}{6982088499743}a^{9}+\frac{414300106212}{6982088499743}a^{8}+\frac{2761116338595}{6982088499743}a^{7}+\frac{1774498338752}{6982088499743}a^{6}-\frac{3114912606773}{6982088499743}a^{5}-\frac{3332481019292}{6982088499743}a^{4}-\frac{2809198037392}{6982088499743}a^{3}+\frac{43265610642}{997441214249}a^{2}-\frac{1374761642938}{6982088499743}a+\frac{1183428980269}{6982088499743}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{11}$, which has order $11$
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: | $\frac{392008010}{6982088499743}a^{9}-\frac{871765612}{997441214249}a^{8}+\frac{2181963622}{997441214249}a^{7}-\frac{1554660095}{997441214249}a^{6}+\frac{77513860604}{6982088499743}a^{5}-\frac{354498115191}{6982088499743}a^{4}+\frac{151431680912}{6982088499743}a^{3}-\frac{782745638481}{6982088499743}a^{2}-\frac{5556339924}{997441214249}a-\frac{4632977307815}{6982088499743}$, $\frac{3546220780}{6982088499743}a^{9}+\frac{2343241681}{997441214249}a^{8}-\frac{5796300724}{997441214249}a^{7}-\frac{29812377274}{997441214249}a^{6}+\frac{128361826960}{6982088499743}a^{5}+\frac{2059002753228}{6982088499743}a^{4}+\frac{3115480695672}{6982088499743}a^{3}+\frac{3458318583870}{6982088499743}a^{2}+\frac{993372536776}{997441214249}a+\frac{20622797485746}{6982088499743}$, $\frac{88478170}{6982088499743}a^{9}+\frac{2463196656}{997441214249}a^{8}-\frac{1473151374}{997441214249}a^{7}-\frac{25703093039}{997441214249}a^{6}-\frac{131462410692}{6982088499743}a^{5}+\frac{1746977099638}{6982088499743}a^{4}+\frac{1823390851032}{6982088499743}a^{3}+\frac{1601744917405}{6982088499743}a^{2}+\frac{816574808300}{997441214249}a+\frac{12180684926460}{6982088499743}$, $\frac{493963230}{997441214249}a^{9}-\frac{119954975}{997441214249}a^{8}-\frac{4323149350}{997441214249}a^{7}-\frac{4109284235}{997441214249}a^{6}+\frac{37117748236}{997441214249}a^{5}+\frac{44575093370}{997441214249}a^{4}+\frac{184584263520}{997441214249}a^{3}+\frac{265224809495}{997441214249}a^{2}+\frac{176797728476}{997441214249}a+\frac{208574865649}{997441214249}$ | 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: | \( 257.113789169 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 257.113789169 \cdot 11}{2\cdot\sqrt{5000000000000000}}\cr\approx \mathstrut & 0.195840449765 \end{aligned}\]
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 | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | R | ${\href{/padicField/7.2.0.1}{2} }^{5}$ | ${\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.10.0.1}{10} }$ | ${\href{/padicField/17.5.0.1}{5} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }$ | ${\href{/padicField/29.10.0.1}{10} }$ | ${\href{/padicField/31.10.0.1}{10} }$ | ${\href{/padicField/37.10.0.1}{10} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.1.0.1}{1} }^{10}$ | ${\href{/padicField/47.10.0.1}{10} }$ | ${\href{/padicField/53.10.0.1}{10} }$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.10.15.9 | $x^{10} + 20 x^{9} + 206 x^{8} + 1376 x^{7} + 6504 x^{6} + 22496 x^{5} + 57328 x^{4} + 105856 x^{3} + 135504 x^{2} + 108864 x + 9568$ | $2$ | $5$ | $15$ | $C_{10}$ | $[3]^{5}$ |
\(5\) | 5.10.16.7 | $x^{10} + 40 x^{9} + 400 x^{8} + 10 x^{5} + 200 x^{4} - 1225$ | $5$ | $2$ | $16$ | $C_{10}$ | $[2]^{2}$ |
Artin representations
Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
---|---|---|---|---|---|---|---|
* | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
* | 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
* | 1.25.5t1.a.b | $1$ | $ 5^{2}$ | 5.5.390625.1 | $C_5$ (as 5T1) | $0$ | $1$ |
* | 1.200.10t1.a.a | $1$ | $ 2^{3} \cdot 5^{2}$ | 10.0.5000000000000000.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ |
* | 1.25.5t1.a.d | $1$ | $ 5^{2}$ | 5.5.390625.1 | $C_5$ (as 5T1) | $0$ | $1$ |
* | 1.200.10t1.a.d | $1$ | $ 2^{3} \cdot 5^{2}$ | 10.0.5000000000000000.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ |
* | 1.25.5t1.a.a | $1$ | $ 5^{2}$ | 5.5.390625.1 | $C_5$ (as 5T1) | $0$ | $1$ |
* | 1.200.10t1.a.c | $1$ | $ 2^{3} \cdot 5^{2}$ | 10.0.5000000000000000.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ |
* | 1.25.5t1.a.c | $1$ | $ 5^{2}$ | 5.5.390625.1 | $C_5$ (as 5T1) | $0$ | $1$ |
* | 1.200.10t1.a.b | $1$ | $ 2^{3} \cdot 5^{2}$ | 10.0.5000000000000000.1 | $C_{10}$ (as 10T1) | $0$ | $-1$ |