Normalized defining polynomial
\( x^{10} - x^{9} + 5x^{8} - 13x^{7} + 17x^{6} - 7x^{5} + 13x^{4} - 54x^{3} + 78x^{2} - 45x + 9 \)
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: | \(-429162451875\) \(\medspace = -\,3^{5}\cdot 5^{4}\cdot 41^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(14.56\) | 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: | $3^{1/2}5^{1/2}41^{1/2}\approx 24.79919353527449$ | ||
Ramified primes: | \(3\), \(5\), \(41\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}$, $\frac{1}{9}a^{8}+\frac{1}{9}a^{7}+\frac{1}{9}a^{6}-\frac{2}{9}a^{5}+\frac{4}{9}a^{4}+\frac{1}{9}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{459}a^{9}+\frac{4}{459}a^{8}+\frac{76}{459}a^{7}-\frac{41}{459}a^{6}+\frac{169}{459}a^{5}-\frac{182}{459}a^{4}+\frac{8}{51}a^{3}-\frac{2}{9}a^{2}+\frac{20}{51}a+\frac{10}{51}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $4$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{1246}{459} a^{9} - \frac{575}{459} a^{8} + \frac{5905}{459} a^{7} - \frac{13040}{459} a^{6} + \frac{14071}{459} a^{5} - \frac{1148}{459} a^{4} + \frac{5203}{153} a^{3} - \frac{1157}{9} a^{2} + \frac{2419}{17} a - \frac{2279}{51} \) (order $6$) | 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: | $\frac{1997}{153}a^{9}-\frac{971}{153}a^{8}+\frac{9499}{153}a^{7}-\frac{21068}{153}a^{6}+\frac{23197}{153}a^{5}-\frac{2102}{153}a^{4}+\frac{24920}{153}a^{3}-\frac{1862}{3}a^{2}+\frac{35738}{51}a-\frac{3898}{17}$, $\frac{205}{153}a^{9}-\frac{98}{153}a^{8}+\frac{994}{153}a^{7}-\frac{2132}{153}a^{6}+\frac{2464}{153}a^{5}-\frac{284}{153}a^{4}+\frac{823}{51}a^{3}-\frac{191}{3}a^{2}+\frac{1261}{17}a-\frac{415}{17}$, $\frac{6674}{459}a^{9}-\frac{3139}{459}a^{8}+\frac{31700}{459}a^{7}-\frac{69991}{459}a^{6}+\frac{76337}{459}a^{5}-\frac{6274}{459}a^{4}+\frac{9277}{51}a^{3}-\frac{6196}{9}a^{2}+\frac{39181}{51}a-\frac{12565}{51}$, $\frac{1472}{459}a^{9}-\frac{589}{459}a^{8}+\frac{7016}{459}a^{7}-\frac{14962}{459}a^{6}+\frac{16004}{459}a^{5}-\frac{970}{459}a^{4}+\frac{6190}{153}a^{3}-\frac{1330}{9}a^{2}+\frac{8258}{51}a-\frac{2569}{51}$ | 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: | \( 533.709602782 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 533.709602782 \cdot 1}{6\cdot\sqrt{429162451875}}\cr\approx \mathstrut & 1.32966503735 \end{aligned}\]
Galois group
A solvable group of order 20 |
The 8 conjugacy class representatives for $D_{10}$ |
Character table for $D_{10}$ |
Intermediate fields
\(\Q(\sqrt{-3}) \), 5.1.378225.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 20.0.7740181734425675064931640625.1 |
Degree 10 sibling: | 10.2.29326100878125.2 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.10.0.1}{10} }$ | R | R | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.10.0.1}{10} }$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{5}$ | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }$ | ${\href{/padicField/29.10.0.1}{10} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.2.0.1}{2} }^{5}$ | ${\href{/padicField/53.2.0.1}{2} }^{5}$ | ${\href{/padicField/59.2.0.1}{2} }^{5}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
3.2.1.2 | $x^{2} + 3$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(5\) | 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(41\) | 41.2.0.1 | $x^{2} + 38 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
41.4.2.1 | $x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
41.4.2.1 | $x^{4} + 1962 x^{3} + 998289 x^{2} + 35245368 x + 7080121$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |