Normalized defining polynomial
\( x^{23} - 8x - 4 \)
Invariants
Degree: | $23$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[3, 10]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(48051614335700752762452449047826247066845184\) \(\medspace = 2^{22}\cdot 3\cdot 2377\cdot 111863\cdot 115741\cdot 198173\cdot 4489367\cdot 139474352947\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(79.29\) | 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: | $2^{22/23}3^{1/2}2377^{1/2}111863^{1/2}115741^{1/2}198173^{1/2}4489367^{1/2}139474352947^{1/2}\approx 6.568493493524864e+18$ | ||
Ramified primes: | \(2\), \(3\), \(2377\), \(111863\), \(115741\), \(198173\), \(4489367\), \(139474352947\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{11456\!\cdots\!13721}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}$, $\frac{1}{2}a^{13}$, $\frac{1}{2}a^{14}$, $\frac{1}{2}a^{15}$, $\frac{1}{2}a^{16}$, $\frac{1}{2}a^{17}$, $\frac{1}{2}a^{18}$, $\frac{1}{2}a^{19}$, $\frac{1}{2}a^{20}$, $\frac{1}{2}a^{21}$, $\frac{1}{2}a^{22}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $12$ | 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: | $a^{12}+3a+1$, $\frac{1}{2}a^{22}-\frac{1}{2}a^{21}+\frac{1}{2}a^{20}-\frac{1}{2}a^{19}+\frac{1}{2}a^{18}-\frac{1}{2}a^{17}+\frac{1}{2}a^{16}-\frac{1}{2}a^{15}+\frac{1}{2}a^{14}-\frac{1}{2}a^{13}+\frac{1}{2}a^{12}-a^{11}+a^{10}-a^{9}+a^{8}-a^{7}+a^{6}-a^{5}+a^{4}-a^{3}+a^{2}-a-1$, $\frac{1}{2}a^{22}-\frac{3}{2}a^{21}+\frac{3}{2}a^{20}-\frac{1}{2}a^{19}-a^{18}+\frac{5}{2}a^{17}-\frac{3}{2}a^{16}-\frac{1}{2}a^{15}+\frac{5}{2}a^{14}-3a^{13}+\frac{3}{2}a^{12}+2a^{11}-4a^{10}+3a^{9}-4a^{7}+6a^{6}-3a^{5}-3a^{4}+6a^{3}-7a^{2}+a+3$, $6a^{22}-\frac{7}{2}a^{21}+\frac{5}{2}a^{20}-2a^{19}+\frac{3}{2}a^{18}-a^{17}+a^{16}-a^{15}+\frac{1}{2}a^{14}+\frac{1}{2}a^{13}-a^{12}+a^{11}-a^{10}+2a^{9}-3a^{8}+3a^{7}-2a^{6}+2a^{5}-2a^{4}+a^{3}+a^{2}-2a-45$, $\frac{3}{2}a^{22}-a^{21}+\frac{1}{2}a^{20}+\frac{1}{2}a^{19}+\frac{1}{2}a^{17}-\frac{1}{2}a^{16}-a^{15}-\frac{1}{2}a^{13}+\frac{3}{2}a^{12}+a^{11}-2a^{8}-a^{7}-a^{6}+a^{5}+3a^{4}+2a^{3}+a^{2}-4a-15$, $3a^{22}-a^{21}-\frac{3}{2}a^{19}+a^{17}+\frac{3}{2}a^{16}-\frac{3}{2}a^{14}-2a^{13}+\frac{1}{2}a^{12}+2a^{11}+2a^{10}-a^{9}-3a^{8}-2a^{7}+2a^{6}+3a^{5}+a^{4}-4a^{3}-4a^{2}+a-17$, $\frac{9}{2}a^{22}-a^{21}+2a^{20}-a^{19}-\frac{3}{2}a^{18}-a^{17}+a^{16}+\frac{5}{2}a^{15}+\frac{1}{2}a^{14}-2a^{13}-3a^{12}+3a^{10}+4a^{9}-a^{8}-5a^{7}-4a^{6}+3a^{5}+7a^{4}+4a^{3}-6a^{2}-9a-39$, $a^{21}+\frac{3}{2}a^{18}+\frac{1}{2}a^{17}-a^{16}-\frac{1}{2}a^{14}-2a^{13}-a^{12}+2a^{9}+2a^{8}+a^{7}+3a^{6}+a^{5}-5a^{4}-2a^{3}-8a-5$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{20}+a^{19}-a^{18}+a^{16}-\frac{1}{2}a^{14}+2a^{13}-3a^{12}-a^{11}-a^{9}+a^{8}+5a^{7}+2a^{5}-2a^{4}-3a^{3}-4a^{2}-a+1$, $3a^{22}+a^{20}-\frac{3}{2}a^{19}-\frac{1}{2}a^{18}+2a^{16}+\frac{3}{2}a^{15}-a^{14}-\frac{3}{2}a^{13}-\frac{3}{2}a^{12}+2a^{11}+3a^{10}-3a^{8}-5a^{7}+a^{6}+5a^{5}+4a^{4}-a^{3}-9a^{2}-4a-19$, $a^{22}+\frac{5}{2}a^{21}+2a^{20}-\frac{1}{2}a^{19}-2a^{18}-4a^{17}-\frac{3}{2}a^{16}+a^{15}+\frac{11}{2}a^{14}+4a^{13}+2a^{12}-5a^{11}-7a^{10}-8a^{9}+a^{8}+8a^{7}+14a^{6}+7a^{5}-2a^{4}-16a^{3}-18a^{2}-11a+3$, $\frac{1}{2}a^{22}-3a^{21}+\frac{1}{2}a^{20}+a^{19}+3a^{18}+2a^{17}-\frac{5}{2}a^{15}-\frac{9}{2}a^{14}-\frac{3}{2}a^{13}+\frac{1}{2}a^{12}+6a^{11}+5a^{10}+3a^{9}-5a^{8}-8a^{7}-8a^{6}-2a^{5}+9a^{4}+12a^{3}+10a^{2}-5a-17$ (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: | \( 13271946732000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{10}\cdot 13271946732000 \cdot 1}{2\cdot\sqrt{48051614335700752762452449047826247066845184}}\cr\approx \mathstrut & 0.734411237387189 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 25852016738884976640000 |
The 1255 conjugacy class representatives for $S_{23}$ |
Character table for $S_{23}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 46 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 | R | $15{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | $20{,}\,{\href{/padicField/7.3.0.1}{3} }$ | $15{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.5.0.1}{5} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.7.0.1}{7} }{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.11.0.1}{11} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.11.0.1}{11} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/29.3.0.1}{3} }$ | ${\href{/padicField/31.10.0.1}{10} }{,}\,{\href{/padicField/31.7.0.1}{7} }{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.12.0.1}{12} }{,}\,{\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | $16{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ | ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.7.0.1}{7} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/59.5.0.1}{5} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.23.22.1 | $x^{23} + 2$ | $23$ | $1$ | $22$ | $C_{23}:C_{11}$ | $[\ ]_{23}^{11}$ |
\(3\) | 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.3.0.1 | $x^{3} + 2 x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
\(2377\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $21$ | $1$ | $21$ | $0$ | $C_{21}$ | $[\ ]^{21}$ | ||
\(111863\) | $\Q_{111863}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{111863}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
\(115741\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
\(198173\) | $\Q_{198173}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{198173}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | ||
\(4489367\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $18$ | $1$ | $18$ | $0$ | $C_{18}$ | $[\ ]^{18}$ | ||
\(139474352947\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |