Normalized defining polynomial
\( x^{23} - 3 \)
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)
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Discriminant: | \(-655251210967044330484283743635659792988303\) \(\medspace = -\,3^{22}\cdot 23^{23}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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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: | $3^{22/23}23^{527/506}\approx 74.92367730844663$ | ||
Ramified primes: | \(3\), \(23\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{-23}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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)
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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: | $a^{12}-a-1$, $a^{22}+a^{21}+a^{20}+a^{19}+a^{17}-a^{14}-a^{13}-a^{12}-2a^{11}-a^{10}-2a^{9}-a^{8}-2a^{7}-a^{6}+a^{3}+3a+2$, $a^{22}+a^{21}+a^{20}+a^{19}+a^{17}+a^{16}-a^{15}-a^{12}-a^{10}-a^{9}-a^{7}+a^{5}+a^{3}+3a^{2}+2a+1$, $a^{22}+a^{15}+a^{13}+a^{12}+a^{11}+a^{10}+a^{9}+a^{8}-a^{7}-a^{5}-a^{4}+a^{2}+2a+1$, $a^{18}+a^{16}-a^{15}+2a^{14}-a^{13}+2a^{12}-2a^{11}+2a^{10}-2a^{9}+2a^{8}-3a^{7}+2a^{6}-3a^{5}+2a^{4}-3a^{3}+a^{2}-2a+1$, $a^{22}+a^{21}-2a^{20}+2a^{16}+a^{15}-a^{14}-a^{13}-2a^{12}-4a^{11}+a^{10}+3a^{7}-2a^{6}-2a^{5}-a^{4}-2a^{3}+a^{2}+6a+2$, $a^{22}+a^{20}-a^{16}+a^{15}-a^{14}+3a^{13}-a^{12}+3a^{11}-3a^{10}+2a^{9}-3a^{8}+3a^{7}-a^{6}+4a^{5}-a^{4}+2a^{3}-3a^{2}+a-2$, $a^{22}+a^{21}-2a^{19}-2a^{18}-2a^{17}+a^{15}+4a^{14}+3a^{13}+2a^{12}-2a^{11}-2a^{10}-4a^{9}-2a^{8}-a^{7}+3a^{6}+2a^{5}+a^{4}-a^{3}+1$, $2a^{21}-2a^{20}+a^{19}+a^{18}-3a^{17}+2a^{16}-a^{15}-2a^{14}+a^{13}-3a^{11}+3a^{10}-a^{9}-2a^{8}+4a^{7}-2a^{6}-a^{5}+5a^{4}-4a^{3}+2a^{2}+3a-2$, $a^{22}-2a^{20}+a^{19}+2a^{17}-2a^{16}-a^{15}+a^{14}+a^{13}+a^{12}-4a^{11}+a^{10}+a^{9}+3a^{8}-2a^{7}-3a^{6}+2a^{5}+a^{4}+3a^{3}-5a^{2}+a+1$, $4a^{22}-a^{21}-5a^{20}-7a^{19}-5a^{18}+3a^{17}+9a^{16}+10a^{15}+5a^{14}-4a^{13}-10a^{12}-9a^{11}-5a^{10}+4a^{9}+13a^{8}+13a^{7}+4a^{6}-11a^{5}-22a^{4}-16a^{3}+a^{2}+14a+20$ (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: | \( 1289649122770 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 1289649122770 \cdot 1}{2\cdot\sqrt{655251210967044330484283743635659792988303}}\cr\approx \mathstrut & 0.959944672896699 \end{aligned}\] (assuming GRH)
Galois group
A solvable group of order 506 |
The 23 conjugacy class representatives for $F_{23}$ |
Character table for $F_{23}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 46 sibling: | deg 46 |
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 | ${\href{/padicField/2.11.0.1}{11} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | R | $22{,}\,{\href{/padicField/5.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.11.0.1}{11} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/19.1.0.1}{1} }$ | R | ${\href{/padicField/29.11.0.1}{11} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.11.0.1}{11} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.11.0.1}{11} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/43.1.0.1}{1} }$ | $23$ | $22{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.11.0.1}{11} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.23.22.1 | $x^{23} + 3$ | $23$ | $1$ | $22$ | $C_{23}:C_{11}$ | $[\ ]_{23}^{11}$ |
\(23\) | 23.23.23.12 | $x^{23} + 23 x + 23$ | $23$ | $1$ | $23$ | $F_{23}$ | $[23/22]_{22}$ |