Normalized defining polynomial
\( x^{20} - 4x + 2 \)
Invariants
Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-2175240440437716006500319000164040704\) \(\medspace = -\,2^{40}\cdot 31\cdot 127\cdot 137\cdot 173\cdot 1073393\cdot 19752251119\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(65.60\) | 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: | not computed | ||
Ramified primes: | \(2\), \(31\), \(127\), \(137\), \(173\), \(1073393\), \(19752251119\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-19783\!\cdots\!23979}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $10$ | 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: | $a-1$, $a^{19}+a^{18}-a^{17}+a^{15}-a^{14}+2a^{12}-a^{11}-a^{10}+2a^{9}-a^{8}-2a^{7}+3a^{6}-3a^{4}+3a^{3}+a^{2}-4a-1$, $9a^{19}+5a^{18}+2a^{17}+2a^{16}+a^{10}-a^{9}-a^{6}+a^{5}-a-37$, $4a^{19}+a^{18}+a^{17}+4a^{16}-a^{14}-3a^{13}+3a^{12}+2a^{11}+a^{10}-6a^{9}+a^{8}+2a^{7}+5a^{6}-4a^{5}-5a^{4}+a^{3}+7a^{2}+2a-23$, $6a^{19}+6a^{18}+6a^{17}-a^{16}-3a^{15}+4a^{14}+3a^{13}-5a^{12}-2a^{11}+7a^{10}+a^{9}-9a^{8}+a^{7}+11a^{6}-3a^{5}-12a^{4}+5a^{3}+14a^{2}-9a-37$, $a^{17}+2a^{16}+2a^{15}+3a^{14}+2a^{13}+3a^{12}+2a^{11}+a^{9}-2a^{8}-3a^{7}-2a^{6}-4a^{5}-5a^{4}-2a^{3}-3a^{2}-3a+3$, $a^{19}+3a^{18}+3a^{16}-a^{15}+3a^{14}-a^{13}+4a^{12}-a^{11}+3a^{10}-a^{9}+4a^{8}+2a^{6}+a^{5}+a^{4}+4a^{3}-a^{2}+6a-9$, $a^{19}-2a^{18}+3a^{17}-4a^{16}+5a^{15}-6a^{14}+7a^{13}-8a^{12}+8a^{11}-8a^{10}+7a^{9}-7a^{8}+6a^{7}-6a^{6}+4a^{5}-3a^{4}+a^{3}-2a+1$, $a^{19}-2a^{17}+5a^{15}+4a^{14}+3a^{13}+4a^{12}-2a^{11}-7a^{10}-4a^{9}-4a^{8}-4a^{7}+6a^{6}+9a^{5}+3a^{4}+6a^{3}+2a^{2}-12a-13$, $5a^{19}+3a^{18}+2a^{17}+a^{16}-a^{15}+a^{13}+a^{12}-2a^{10}+a^{9}+a^{8}+a^{7}-a^{6}-3a^{5}+3a^{4}+3a^{2}-5a-19$ (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: | \( 34634743019.0 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{9}\cdot 34634743019.0 \cdot 1}{2\cdot\sqrt{2175240440437716006500319000164040704}}\cr\approx \mathstrut & 0.716815101340 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 2432902008176640000 |
The 627 conjugacy class representatives for $S_{20}$ |
Character table for $S_{20}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 40 sibling: | data not computed |
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 | R | $15{,}\,{\href{/padicField/3.5.0.1}{5} }$ | ${\href{/padicField/5.10.0.1}{10} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.13.0.1}{13} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.7.0.1}{7} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.9.0.1}{9} }{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.11.0.1}{11} }{,}\,{\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $20$ | ${\href{/padicField/23.13.0.1}{13} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ | $15{,}\,{\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.9.0.1}{9} }{,}\,{\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | $18{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | $20$ | ${\href{/padicField/59.10.0.1}{10} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | Deg $20$ | $20$ | $1$ | $40$ | |||
\(31\) | 31.2.0.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
31.2.1.2 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
31.5.0.1 | $x^{5} + 7 x + 28$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
31.11.0.1 | $x^{11} + 20 x + 28$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | |
\(127\) | $\Q_{127}$ | $x + 124$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
127.2.1.1 | $x^{2} + 381$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
127.3.0.1 | $x^{3} + 3 x + 124$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
127.3.0.1 | $x^{3} + 3 x + 124$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
127.11.0.1 | $x^{11} + 11 x + 124$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | |
\(137\) | $\Q_{137}$ | $x + 134$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
137.2.1.2 | $x^{2} + 411$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
137.6.0.1 | $x^{6} + x^{4} + 116 x^{3} + 102 x^{2} + 3 x + 3$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
137.11.0.1 | $x^{11} + x + 134$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | |
\(173\) | $\Q_{173}$ | $x + 171$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{173}$ | $x + 171$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
173.2.1.2 | $x^{2} + 346$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
173.7.0.1 | $x^{7} + 5 x + 171$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
173.9.0.1 | $x^{9} + 56 x^{2} + 104 x + 171$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | |
\(1073393\) | $\Q_{1073393}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{1073393}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{1073393}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $12$ | $1$ | $12$ | $0$ | $C_{12}$ | $[\ ]^{12}$ | ||
\(19752251119\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ |