Normalized defining polynomial
\( x^{10} - 3x^{9} + 2x^{8} + 15x^{7} - 29x^{6} + 30x^{5} - 17x^{4} + 21x^{3} - 4x^{2} - 15x - 5 \)
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: | \(2099520000000\) \(\medspace = 2^{12}\cdot 3^{8}\cdot 5^{7}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(17.07\) | 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: | $2^{3/2}3^{4/5}5^{3/4}\approx 22.775582886985283$ | ||
Ramified primes: | \(2\), \(3\), \(5\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{10}a^{8}+\frac{1}{10}a^{6}-\frac{1}{5}a^{5}+\frac{2}{5}a^{4}+\frac{2}{5}a^{3}+\frac{1}{10}a^{2}-\frac{1}{2}$, $\frac{1}{43930}a^{9}+\frac{1797}{43930}a^{8}-\frac{3039}{43930}a^{7}-\frac{90}{4393}a^{6}+\frac{1856}{4393}a^{5}-\frac{2564}{21965}a^{4}+\frac{3669}{43930}a^{3}-\frac{20423}{43930}a^{2}+\frac{4237}{8786}a-\frac{2022}{4393}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
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: | $\frac{3251}{43930}a^{9}-\frac{9429}{43930}a^{8}+\frac{4461}{43930}a^{7}+\frac{26277}{21965}a^{6}-\frac{45774}{21965}a^{5}+\frac{37494}{21965}a^{4}-\frac{2451}{8786}a^{3}+\frac{18201}{43930}a^{2}+\frac{6825}{8786}a-\frac{1594}{4393}$, $\frac{1}{382}a^{9}+\frac{4}{955}a^{8}+\frac{17}{382}a^{7}-\frac{107}{1910}a^{6}-\frac{13}{955}a^{5}+\frac{741}{955}a^{4}-\frac{373}{1910}a^{3}-\frac{156}{955}a^{2}-\frac{207}{382}a-\frac{165}{382}$, $\frac{2276}{21965}a^{9}-\frac{13001}{43930}a^{8}+\frac{2211}{21965}a^{7}+\frac{7655}{4393}a^{6}-\frac{12406}{4393}a^{5}+\frac{36017}{21965}a^{4}+\frac{3944}{21965}a^{3}+\frac{56279}{43930}a^{2}-\frac{8009}{4393}a-\frac{809}{4393}$, $\frac{725}{4393}a^{9}-\frac{32139}{43930}a^{8}+\frac{8415}{8786}a^{7}+\frac{95251}{43930}a^{6}-\frac{183301}{21965}a^{5}+\frac{230612}{21965}a^{4}-\frac{190778}{21965}a^{3}+\frac{271681}{43930}a^{2}-\frac{54715}{8786}a-\frac{39655}{8786}$, $\frac{15007}{43930}a^{9}-\frac{26882}{21965}a^{8}+\frac{29481}{21965}a^{7}+\frac{195467}{43930}a^{6}-\frac{274292}{21965}a^{5}+\frac{369351}{21965}a^{4}-\frac{660129}{43930}a^{3}+\frac{333103}{21965}a^{2}-\frac{37152}{4393}a-\frac{16585}{8786}$ | 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: | \( 392.249740018 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 392.249740018 \cdot 1}{2\cdot\sqrt{2099520000000}}\cr\approx \mathstrut & 0.843824489755 \end{aligned}\]
Galois group
A solvable group of order 20 |
The 5 conjugacy class representatives for $F_5$ |
Character table for $F_5$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 5.1.648000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Galois closure: | 20.0.1410554953728000000000000000.1 |
Degree 5 sibling: | 5.1.648000.1 |
Minimal sibling: | 5.1.648000.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | R | R | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.5.0.1}{5} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
2.4.6.3 | $x^{4} + 8 x^{3} + 28 x^{2} + 48 x + 84$ | $2$ | $2$ | $6$ | $C_4$ | $[3]^{2}$ | |
2.4.6.3 | $x^{4} + 8 x^{3} + 28 x^{2} + 48 x + 84$ | $2$ | $2$ | $6$ | $C_4$ | $[3]^{2}$ | |
\(3\) | 3.10.8.1 | $x^{10} + 10 x^{9} + 50 x^{8} + 160 x^{7} + 360 x^{6} + 598 x^{5} + 750 x^{4} + 640 x^{3} + 280 x^{2} + 40 x + 17$ | $5$ | $2$ | $8$ | $F_5$ | $[\ ]_{5}^{4}$ |
\(5\) | 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ |