Normalized defining polynomial
\( x^{10} - x^{9} - x^{8} - 4x^{7} + 18x^{6} + x^{5} - 8x^{4} + 3x^{3} + 6x^{2} - 3x - 1 \)
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: | \(626668503125\) \(\medspace = 5^{5}\cdot 7^{4}\cdot 17^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(15.13\) | 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: | $5^{1/2}7^{1/2}17^{1/2}\approx 24.392621835300936$ | ||
Ramified primes: | \(5\), \(7\), \(17\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{23711}a^{9}-\frac{11085}{23711}a^{8}-\frac{4263}{23711}a^{7}-\frac{4935}{23711}a^{6}-\frac{1719}{23711}a^{5}-\frac{10247}{23711}a^{4}+\frac{2050}{23711}a^{3}-\frac{39}{131}a^{2}-\frac{4338}{23711}a-\frac{3519}{23711}$
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: | $\frac{5675}{23711}a^{9}-\frac{2092}{23711}a^{8}-\frac{7305}{23711}a^{7}-\frac{27145}{23711}a^{6}+\frac{84740}{23711}a^{5}+\frac{58780}{23711}a^{4}-\frac{8351}{23711}a^{3}+\frac{65}{131}a^{2}+\frac{41290}{23711}a-\frac{5663}{23711}$, $\frac{5663}{23711}a^{9}-\frac{11338}{23711}a^{8}-\frac{3571}{23711}a^{7}-\frac{15347}{23711}a^{6}+\frac{129079}{23711}a^{5}-\frac{79077}{23711}a^{4}-\frac{104084}{23711}a^{3}+\frac{140}{131}a^{2}+\frac{22213}{23711}a-\frac{34568}{23711}$, $\frac{3481}{23711}a^{9}-\frac{9088}{23711}a^{8}+\frac{3583}{23711}a^{7}-\frac{11971}{23711}a^{6}+\frac{86177}{23711}a^{5}-\frac{103307}{23711}a^{4}-\frac{961}{23711}a^{3}+\frac{88}{131}a^{2}+\frac{3329}{23711}a-\frac{38474}{23711}$, $\frac{15854}{23711}a^{9}-\frac{19369}{23711}a^{8}-\frac{9252}{23711}a^{7}-\frac{64323}{23711}a^{6}+\frac{299156}{23711}a^{5}-\frac{59299}{23711}a^{4}-\frac{78214}{23711}a^{3}+\frac{407}{131}a^{2}+\frac{58381}{23711}a-\frac{45665}{23711}$, $\frac{25239}{23711}a^{9}-\frac{31937}{23711}a^{8}-\frac{17050}{23711}a^{7}-\frac{95426}{23711}a^{6}+\frac{479509}{23711}a^{5}-\frac{103000}{23711}a^{4}-\frac{187140}{23711}a^{3}+\frac{668}{131}a^{2}+\frac{129171}{23711}a-\frac{113190}{23711}$ | 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: | \( 57.6899187766 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 57.6899187766 \cdot 1}{2\cdot\sqrt{626668503125}}\cr\approx \mathstrut & 0.227159269273 \end{aligned}\]
Galois group
A solvable group of order 20 |
The 8 conjugacy class representatives for $D_{10}$ |
Character table for $D_{10}$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 5.1.14161.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 20.0.5561214638787231316416015625.1 |
Degree 10 sibling: | 10.0.74573551871875.1 |
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 | ${\href{/padicField/2.10.0.1}{10} }$ | ${\href{/padicField/3.10.0.1}{10} }$ | R | R | ${\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{5}$ | R | ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{5}$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{5}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.10.0.1}{10} }$ | ${\href{/padicField/47.2.0.1}{2} }^{5}$ | ${\href{/padicField/53.10.0.1}{10} }$ | ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.10.5.1 | $x^{10} + 100 x^{9} + 4025 x^{8} + 82000 x^{7} + 860258 x^{6} + 4015486 x^{5} + 4317350 x^{4} + 2373700 x^{3} + 3853141 x^{2} + 15123594 x + 12051954$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
\(7\) | 7.2.0.1 | $x^{2} + 6 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
7.4.2.1 | $x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
\(17\) | 17.2.0.1 | $x^{2} + 16 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
17.4.2.1 | $x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |