Normalized defining polynomial
\( x^{22} - 4x - 1 \)
Invariants
Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[2, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(24505586101645802782745775444611305\) \(\medspace = 5\cdot 549569\cdot 7345029585473\cdot 1214169483036053\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(36.57\) | 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: | $5^{1/2}549569^{1/2}7345029585473^{1/2}1214169483036053^{1/2}\approx 1.5654260155512237e+17$ | ||
Ramified primes: | \(5\), \(549569\), \(7345029585473\), \(1214169483036053\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{24505\!\cdots\!11305}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{7}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{8}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{9}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{10}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
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: | $\frac{1}{2}a^{21}+\frac{1}{2}a^{10}-2$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{10}-2$, $\frac{1}{2}a^{21}+\frac{1}{2}a^{10}+a^{9}+a^{8}+a^{7}+a^{6}+a^{5}+a^{4}+a^{3}+a^{2}+a-1$, $\frac{1}{2}a^{19}+\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{14}+\frac{1}{2}a^{13}+\frac{1}{2}a^{11}-\frac{1}{2}a^{8}+\frac{1}{2}a^{6}+\frac{1}{2}a^{5}+a^{4}+\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-a+\frac{1}{2}$, $\frac{1}{2}a^{20}+\frac{1}{2}a^{19}-\frac{1}{2}a^{18}+\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}+a^{13}+\frac{1}{2}a^{12}-\frac{1}{2}a^{11}+\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}+a^{6}+\frac{3}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+a^{2}-\frac{1}{2}a-\frac{3}{2}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{19}-\frac{1}{2}a^{15}+\frac{1}{2}a^{14}-\frac{1}{2}a^{13}-\frac{1}{2}a^{9}+\frac{1}{2}a^{8}-a^{6}+a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}+\frac{1}{2}a^{2}-2$, $\frac{1}{2}a^{21}+\frac{1}{2}a^{18}+\frac{1}{2}a^{13}+\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-a^{6}-a^{4}-a^{3}-\frac{1}{2}a^{2}-1$, $\frac{1}{2}a^{20}+\frac{1}{2}a^{19}-a^{14}-\frac{1}{2}a^{13}+\frac{1}{2}a^{11}+\frac{1}{2}a^{9}+\frac{1}{2}a^{8}+a^{7}-a^{5}-a^{4}-\frac{1}{2}a^{2}-a+\frac{1}{2}$, $\frac{1}{2}a^{21}+\frac{1}{2}a^{19}+\frac{1}{2}a^{13}+\frac{1}{2}a^{12}+\frac{1}{2}a^{11}+\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-a^{7}-a^{6}-2a^{5}-a^{4}-a^{3}-\frac{1}{2}a^{2}+\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{20}+\frac{1}{2}a^{19}+\frac{1}{2}a^{18}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}+\frac{1}{2}a^{13}+\frac{1}{2}a^{12}+\frac{1}{2}a^{9}+\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-2a^{6}-\frac{5}{2}a^{5}-\frac{3}{2}a^{4}+\frac{1}{2}a^{2}+\frac{1}{2}a+1$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{19}-\frac{1}{2}a^{18}+\frac{1}{2}a^{17}-a^{15}+\frac{1}{2}a^{13}-a^{12}-a^{11}+a^{10}+\frac{1}{2}a^{9}-\frac{3}{2}a^{8}+\frac{1}{2}a^{7}+\frac{5}{2}a^{6}-a^{4}+2a^{3}+\frac{3}{2}a^{2}-2a$ (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: | \( 364755463.699 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{10}\cdot 364755463.699 \cdot 1}{2\cdot\sqrt{24505586101645802782745775444611305}}\cr\approx \mathstrut & 0.446887223308 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 1124000727777607680000 |
The 1002 conjugacy class representatives for $S_{22}$ |
Character table for $S_{22}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 44 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 | $20{,}\,{\href{/padicField/2.2.0.1}{2} }$ | $17{,}\,{\href{/padicField/3.5.0.1}{5} }$ | R | $17{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.14.0.1}{14} }{,}\,{\href{/padicField/19.8.0.1}{8} }$ | ${\href{/padicField/23.7.0.1}{7} }^{2}{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{3}$ | $16{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | $15{,}\,{\href{/padicField/41.7.0.1}{7} }$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/53.2.0.1}{2} }$ | $16{,}\,{\href{/padicField/59.5.0.1}{5} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.2.0.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
5.11.0.1 | $x^{11} + 3 x + 3$ | $1$ | $11$ | $0$ | $C_{11}$ | $[\ ]^{11}$ | |
\(549569\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(7345029585473\) | $\Q_{7345029585473}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
\(1214169483036053\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ |