Normalized defining polynomial
\( x^{20} - 4x^{15} - 183x^{10} + 104x^{5} + 1 \)
Invariants
Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[4, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(19102496095001697540283203125\) \(\medspace = 5^{30}\cdot 29^{5}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(25.95\) | 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: | $5^{31/20}29^{1/2}\approx 65.25330389102201$ | ||
Ramified primes: | \(5\), \(29\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{29}) \) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}-\frac{1}{3}$, $\frac{1}{3}a^{6}-\frac{1}{3}a$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{3}$, $\frac{1}{3}a^{9}-\frac{1}{3}a^{4}$, $\frac{1}{18}a^{10}+\frac{1}{18}a^{5}+\frac{7}{18}$, $\frac{1}{18}a^{11}+\frac{1}{18}a^{6}+\frac{7}{18}a$, $\frac{1}{18}a^{12}+\frac{1}{18}a^{7}+\frac{7}{18}a^{2}$, $\frac{1}{18}a^{13}+\frac{1}{18}a^{8}+\frac{7}{18}a^{3}$, $\frac{1}{18}a^{14}+\frac{1}{18}a^{9}+\frac{7}{18}a^{4}$, $\frac{1}{378}a^{15}-\frac{1}{42}a^{10}+\frac{17}{126}a^{5}+\frac{19}{189}$, $\frac{1}{378}a^{16}-\frac{1}{42}a^{11}+\frac{17}{126}a^{6}+\frac{19}{189}a$, $\frac{1}{378}a^{17}-\frac{1}{42}a^{12}+\frac{17}{126}a^{7}+\frac{19}{189}a^{2}$, $\frac{1}{378}a^{18}-\frac{1}{42}a^{13}+\frac{17}{126}a^{8}+\frac{19}{189}a^{3}$, $\frac{1}{378}a^{19}-\frac{1}{42}a^{14}+\frac{17}{126}a^{9}+\frac{19}{189}a^{4}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $11$ | 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: | $\frac{29}{378}a^{17}-\frac{19}{63}a^{12}-\frac{884}{63}a^{7}+\frac{2635}{378}a^{2}$, $\frac{2}{189}a^{15}-\frac{5}{126}a^{10}-\frac{29}{14}a^{5}+\frac{173}{378}$, $\frac{1}{126}a^{16}-\frac{1}{63}a^{11}-\frac{97}{63}a^{6}-\frac{97}{42}a$, $\frac{4}{27}a^{19}-\frac{47}{378}a^{18}-\frac{2}{63}a^{16}+\frac{4}{189}a^{15}-\frac{11}{18}a^{14}+\frac{32}{63}a^{13}+\frac{5}{42}a^{11}-\frac{5}{63}a^{10}-\frac{487}{18}a^{9}+\frac{159}{7}a^{8}+\frac{247}{42}a^{6}-\frac{80}{21}a^{5}+\frac{1027}{54}a^{4}-\frac{5545}{378}a^{3}-\frac{257}{126}a+\frac{110}{189}$, $\frac{44}{189}a^{18}-\frac{1}{42}a^{16}-\frac{13}{14}a^{13}+\frac{13}{126}a^{11}-\frac{5371}{126}a^{8}+\frac{547}{126}a^{6}+\frac{8951}{378}a^{3}-\frac{211}{63}a$, $\frac{47}{378}a^{18}+\frac{1}{42}a^{17}+\frac{1}{42}a^{16}-\frac{32}{63}a^{13}-\frac{13}{126}a^{12}-\frac{13}{126}a^{11}-\frac{159}{7}a^{8}-\frac{547}{126}a^{7}-\frac{547}{126}a^{6}+\frac{5545}{378}a^{3}+\frac{274}{63}a^{2}+\frac{211}{63}a$, $\frac{13}{42}a^{19}+\frac{1}{42}a^{17}-\frac{155}{126}a^{14}-\frac{13}{126}a^{12}-\frac{7139}{126}a^{9}-\frac{547}{126}a^{7}+\frac{1931}{63}a^{4}+\frac{274}{63}a^{2}$, $\frac{29}{378}a^{18}-\frac{2}{63}a^{16}-\frac{19}{63}a^{13}+\frac{5}{42}a^{11}-\frac{884}{63}a^{8}+\frac{247}{42}a^{6}+\frac{2635}{378}a^{3}-\frac{257}{126}a+1$, $\frac{1}{3}a^{19}+\frac{1}{18}a^{18}+\frac{2}{189}a^{17}-\frac{5}{378}a^{16}-\frac{1}{378}a^{15}-\frac{4}{3}a^{14}-\frac{2}{9}a^{13}-\frac{5}{126}a^{12}+\frac{4}{63}a^{11}+\frac{1}{42}a^{10}-61a^{9}-\frac{92}{9}a^{8}-\frac{29}{14}a^{7}+\frac{143}{63}a^{6}+\frac{67}{126}a^{5}+35a^{4}+\frac{115}{18}a^{3}+\frac{551}{378}a^{2}-\frac{715}{378}a-\frac{145}{189}$, $\frac{16}{189}a^{19}+\frac{16}{189}a^{18}+\frac{1}{126}a^{17}+\frac{2}{63}a^{16}+\frac{2}{63}a^{15}-\frac{20}{63}a^{14}-\frac{20}{63}a^{13}-\frac{1}{63}a^{12}-\frac{5}{42}a^{11}-\frac{5}{42}a^{10}-\frac{109}{7}a^{9}-\frac{109}{7}a^{8}-\frac{97}{63}a^{7}-\frac{247}{42}a^{6}-\frac{247}{42}a^{5}+\frac{881}{189}a^{4}+\frac{881}{189}a^{3}-\frac{97}{42}a^{2}+\frac{131}{126}a+\frac{131}{126}$, $\frac{299}{189}a^{19}+\frac{29}{42}a^{18}+\frac{47}{378}a^{17}+\frac{2}{63}a^{16}-\frac{400}{63}a^{14}-\frac{349}{126}a^{13}-\frac{32}{63}a^{12}-\frac{5}{42}a^{11}-\frac{2026}{7}a^{9}-\frac{15919}{126}a^{8}-\frac{159}{7}a^{7}-\frac{247}{42}a^{6}+\frac{31816}{189}a^{4}+\frac{4621}{63}a^{3}+\frac{5545}{378}a^{2}+\frac{257}{126}a-1$ (assuming GRH) | 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: | \( 1748237.43405 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 1748237.43405 \cdot 1}{2\cdot\sqrt{19102496095001697540283203125}}\cr\approx \mathstrut & 0.245801364473 \end{aligned}\] (assuming GRH)
Galois group
$D_{10}:C_4$ (as 20T22):
A solvable group of order 80 |
The 14 conjugacy class representatives for $D_{10}:C_4$ |
Character table for $D_{10}:C_4$ |
Intermediate fields
\(\Q(\sqrt{5}) \), 4.4.725.1, 5.1.78125.1, 10.2.30517578125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 20 siblings: | data not computed |
Degree 40 siblings: | data not computed |
Minimal sibling: | 20.4.78362828735299594700336456298828125.5 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{5}$ | ${\href{/padicField/3.4.0.1}{4} }^{5}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{4}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.10.0.1}{10} }{,}\,{\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{5}$ | ${\href{/padicField/19.2.0.1}{2} }^{9}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{5}$ | ${\href{/padicField/41.10.0.1}{10} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | ${\href{/padicField/47.4.0.1}{4} }^{5}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.2.0.1}{2} }^{10}$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | Deg $20$ | $10$ | $2$ | $30$ | |||
\(29\) | 29.2.1.2 | $x^{2} + 58$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.2.0.1 | $x^{2} + 24 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
29.4.2.1 | $x^{4} + 1440 x^{3} + 535166 x^{2} + 12071520 x + 1504089$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |