Normalized defining polynomial
\( x^{10} - 4x^{9} + 4x^{8} - 4x^{7} + 21x^{6} - 28x^{5} + 8x^{4} - 26x^{3} + 11x^{2} - 13x + 1 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(55730836701\) \(\medspace = 3^{5}\cdot 47^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(11.87\) | 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: | $3^{1/2}47^{1/2}\approx 11.874342087037917$ | ||
Ramified primes: | \(3\), \(47\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{141}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{25}a^{8}-\frac{1}{5}a^{7}-\frac{2}{5}a^{6}+\frac{1}{25}a^{5}+\frac{2}{5}a^{4}-\frac{7}{25}a^{3}+\frac{7}{25}a+\frac{4}{25}$, $\frac{1}{34375}a^{9}-\frac{467}{34375}a^{8}+\frac{399}{1375}a^{7}-\frac{12179}{34375}a^{6}+\frac{1398}{34375}a^{5}+\frac{5823}{34375}a^{4}-\frac{14791}{34375}a^{3}+\frac{7582}{34375}a^{2}-\frac{841}{6875}a-\frac{12473}{34375}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | 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: | $\frac{4836}{34375}a^{9}-\frac{17162}{34375}a^{8}+\frac{439}{1375}a^{7}-\frac{13269}{34375}a^{6}+\frac{98853}{34375}a^{5}-\frac{96222}{34375}a^{4}-\frac{8651}{34375}a^{3}-\frac{149073}{34375}a^{2}+\frac{5674}{6875}a-\frac{32553}{34375}$, $\frac{2986}{34375}a^{9}-\frac{11212}{34375}a^{8}+\frac{389}{1375}a^{7}-\frac{11494}{34375}a^{6}+\frac{57678}{34375}a^{5}-\frac{61272}{34375}a^{4}+\frac{16949}{34375}a^{3}-\frac{82023}{34375}a^{2}-\frac{4051}{6875}a-\frac{17628}{34375}$, $\frac{3138}{34375}a^{9}-\frac{13446}{34375}a^{8}+\frac{537}{1375}a^{7}-\frac{6452}{34375}a^{6}+\frac{63924}{34375}a^{5}-\frac{104301}{34375}a^{4}+\frac{3092}{34375}a^{3}-\frac{29559}{34375}a^{2}+\frac{19367}{6875}a+\frac{11476}{34375}$, $\frac{804}{34375}a^{9}-\frac{2843}{34375}a^{8}+\frac{146}{1375}a^{7}-\frac{8791}{34375}a^{6}+\frac{18492}{34375}a^{5}-\frac{13933}{34375}a^{4}+\frac{40286}{34375}a^{3}-\frac{57197}{34375}a^{2}-\frac{10114}{6875}a-\frac{81542}{34375}$, $\frac{6147}{34375}a^{9}-\frac{24399}{34375}a^{8}+\frac{1028}{1375}a^{7}-\frac{29938}{34375}a^{6}+\frac{130381}{34375}a^{5}-\frac{162269}{34375}a^{4}+\frac{49723}{34375}a^{3}-\frac{177821}{34375}a^{2}+\frac{11373}{6875}a-\frac{77156}{34375}$ | 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: | \( 25.719737423 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 25.719737423 \cdot 1}{2\cdot\sqrt{55730836701}}\cr\approx \mathstrut & 0.33960041942 \end{aligned}\]
Galois group
A solvable group of order 20 |
The 8 conjugacy class representatives for $D_{10}$ |
Character table for $D_{10}$ |
Intermediate fields
\(\Q(\sqrt{141}) \), 5.1.2209.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 20.0.3105926159393528563401.1 |
Degree 10 sibling: | 10.0.1185762483.1 |
Minimal sibling: | 10.0.1185762483.1 |
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 | ${\href{/padicField/5.2.0.1}{2} }^{4}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{5}$ | ${\href{/padicField/17.10.0.1}{10} }$ | ${\href{/padicField/19.2.0.1}{2} }^{5}$ | ${\href{/padicField/23.2.0.1}{2} }^{4}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{5}$ | ${\href{/padicField/37.5.0.1}{5} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{5}$ | R | ${\href{/padicField/53.10.0.1}{10} }$ | ${\href{/padicField/59.10.0.1}{10} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.10.5.2 | $x^{10} + 15 x^{8} + 94 x^{6} + 2 x^{5} + 210 x^{4} - 60 x^{3} + 229 x^{2} + 94 x + 364$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
\(47\) | 47.2.1.1 | $x^{2} + 235$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
47.4.2.1 | $x^{4} + 90 x^{3} + 2129 x^{2} + 4680 x + 96939$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |