Normalized defining polynomial
\( x^{23} + x - 3 \)
Invariants
Degree: | $23$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[1, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-655251210967385758361647963193056439711887\) \(\medspace = -\,859\cdot 1733\cdot 44\!\cdots\!21\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(65.78\) | 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: | $859^{1/2}1733^{1/2}440165607405506986116687141540644921^{1/2}\approx 8.094758865879735e+20$ | ||
Ramified primes: | \(859\), \(1733\), \(44016\!\cdots\!44921\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-65525\!\cdots\!11887}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$
Monogenic: | Yes | |
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: | $a-1$, $a^{21}-a^{19}+a^{18}+a^{17}-a^{16}+a^{14}+a^{9}-a^{7}+a^{6}+a^{5}-a^{4}+a^{2}+a-1$, $a^{21}+a^{19}-a^{15}-a^{13}+a^{9}+a^{7}-a^{3}-a+1$, $2a^{22}+2a^{21}-a^{20}-3a^{19}-2a^{18}+a^{17}+3a^{16}+2a^{15}-2a^{14}-4a^{13}-2a^{12}+2a^{11}+4a^{10}+2a^{9}-3a^{8}-6a^{7}-a^{6}+3a^{5}+5a^{4}+2a^{3}-5a^{2}-7a+2$, $a^{22}+3a^{21}+2a^{20}-a^{19}-2a^{18}+a^{16}-a^{15}-3a^{14}-a^{13}+3a^{12}+4a^{11}+a^{10}-a^{9}+a^{8}+3a^{7}+a^{6}-4a^{5}-5a^{4}+4a^{2}+a-2$, $a^{22}+a^{20}-a^{19}-a^{18}-2a^{16}+a^{15}+a^{14}+2a^{12}-a^{11}-a^{9}-2a^{8}+2a^{7}-2a^{6}+a^{5}+2a^{4}-3a^{3}+2a^{2}-a+1$, $2a^{20}-a^{19}+a^{17}+a^{16}+a^{15}+a^{13}-3a^{10}+a^{9}+a^{8}-3a^{7}+a^{6}-a^{5}-3a^{3}-a^{2}+2$, $4a^{22}-5a^{21}+5a^{20}-5a^{19}+4a^{18}-a^{17}-3a^{16}+6a^{15}-8a^{14}+9a^{13}-8a^{12}+5a^{11}-3a^{10}+5a^{8}-10a^{7}+13a^{6}-15a^{5}+14a^{4}-8a^{3}+6a-8$, $a^{22}+2a^{21}-a^{20}-2a^{19}-2a^{18}+a^{15}+2a^{14}+a^{13}-2a^{12}-3a^{11}-a^{10}-a^{9}+a^{7}+5a^{6}-3a^{4}-5a^{3}+a^{2}-a+2$, $3a^{22}+4a^{21}-5a^{20}-a^{19}+4a^{18}-6a^{17}-5a^{16}+7a^{15}-7a^{13}+5a^{12}+3a^{11}-11a^{10}+8a^{8}-9a^{7}-5a^{6}+13a^{5}-3a^{4}-13a^{3}+9a^{2}+2a-16$, $3a^{22}-4a^{21}+a^{20}-5a^{19}+3a^{18}-4a^{17}+5a^{16}-4a^{15}+6a^{14}-3a^{13}+4a^{12}-6a^{11}+4a^{10}-2a^{9}+3a^{8}-3a^{7}-a^{6}-2a^{5}-2a^{4}+4a^{3}-4a^{2}+8a+1$ (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: | \( 813269772507 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 813269772507 \cdot 1}{2\cdot\sqrt{655251210967385758361647963193056439711887}}\cr\approx \mathstrut & 0.605353791168385 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 25852016738884976640000 |
The 1255 conjugacy class representatives for $S_{23}$ are not computed |
Character table for $S_{23}$ is not computed |
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 | ${\href{/padicField/2.13.0.1}{13} }{,}\,{\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | ${\href{/padicField/3.10.0.1}{10} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.12.0.1}{12} }{,}\,{\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $19{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | $22{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.11.0.1}{11} }{,}\,{\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | $19{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.2.0.1}{2} }^{11}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.10.0.1}{10} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/37.2.0.1}{2} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.12.0.1}{12} }{,}\,{\href{/padicField/53.11.0.1}{11} }$ | $19{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
---|---|---|---|---|---|---|---|
\(859\) | $\Q_{859}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $20$ | $1$ | $20$ | $0$ | 20T1 | $[\ ]^{20}$ | ||
\(1733\) | $\Q_{1733}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{1733}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{1733}$ | $x$ | $1$ | $1$ | $0$ | 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 $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(440\!\cdots\!921\) | $\Q_{44\!\cdots\!21}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{44\!\cdots\!21}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ |