Normalized defining polynomial
\( x^{23} + 4x - 3 \)
Invariants
Degree: | $23$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(-24681102168332647528758926939274761787816079\) \(\medspace = -\,114089\cdot 350671093\cdot 18936502114367\cdot 32577747177637381\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(77.02\) | 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: | $114089^{1/2}350671093^{1/2}18936502114367^{1/2}32577747177637381^{1/2}\approx 4.968007867177008e+21$ | ||
Ramified primes: | \(114089\), \(350671093\), \(18936502114367\), \(32577747177637381\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{-24681\!\cdots\!16079}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | $a^{6}+a^{5}-a^{3}-a^{2}+1$, $a^{16}+a^{9}-a^{8}+a^{2}-2a+1$, $a^{22}+3a^{21}-2a^{19}+2a^{18}+2a^{17}-a^{16}+4a^{15}+8a^{14}+3a^{13}+4a^{12}+10a^{11}+4a^{10}-2a^{9}+4a^{8}+a^{7}-10a^{6}-4a^{5}+2a^{4}-10a^{3}-8a^{2}+8a+4$, $2a^{22}+a^{21}-2a^{20}+a^{19}-a^{18}-2a^{17}+4a^{16}+a^{15}-2a^{14}+3a^{13}-2a^{12}-3a^{11}+4a^{10}-4a^{9}-2a^{8}+8a^{7}-2a^{6}-a^{5}+5a^{4}-9a^{3}-a^{2}+9a-2$, $2a^{22}+3a^{21}+3a^{20}+3a^{19}+3a^{18}+3a^{17}+4a^{16}+5a^{15}+5a^{14}+4a^{13}+3a^{12}+3a^{11}+3a^{10}+3a^{9}+3a^{8}+a^{7}-2a^{6}-2a^{5}-a^{4}-2a^{3}-a^{2}-2a+2$, $a^{22}+a^{21}-a^{20}-a^{19}+a^{18}-a^{16}-a^{13}+a^{12}+2a^{11}-2a^{10}-a^{9}+3a^{8}+a^{7}-3a^{6}+a^{5}-2a^{3}+a^{2}+3a-2$, $2a^{22}+a^{21}+a^{20}+2a^{18}+2a^{17}+2a^{16}+a^{14}+2a^{13}+3a^{12}+a^{11}-a^{10}+a^{8}+2a^{7}-2a^{6}-2a^{5}-2a^{4}+3a^{3}-a^{2}-a+1$, $6a^{22}-5a^{20}-3a^{19}+8a^{18}-7a^{16}-2a^{15}+10a^{14}-11a^{12}+12a^{10}+a^{9}-17a^{8}+4a^{7}+13a^{6}+2a^{5}-25a^{4}+11a^{3}+15a^{2}+a-10$, $4a^{22}-2a^{21}+2a^{20}+4a^{19}-4a^{18}+4a^{17}+2a^{16}-4a^{15}+3a^{14}+4a^{13}-8a^{12}+9a^{11}-a^{10}-5a^{9}+7a^{8}+2a^{7}-11a^{6}+15a^{5}-7a^{4}-8a^{3}+14a^{2}-8a+7$, $15a^{22}+10a^{21}+8a^{20}+10a^{19}+10a^{18}+3a^{17}-6a^{16}-5a^{15}+7a^{14}+15a^{13}+6a^{12}-12a^{11}-15a^{10}+a^{9}+17a^{8}+11a^{7}-8a^{6}-16a^{5}-5a^{4}+9a^{3}+6a^{2}-2a+58$, $a^{22}+2a^{21}-14a^{20}+21a^{19}-10a^{18}-2a^{17}-9a^{16}+4a^{15}+15a^{14}-34a^{13}+20a^{12}-7a^{11}+24a^{10}-26a^{9}-a^{8}+40a^{7}-29a^{6}+16a^{5}-47a^{4}+67a^{3}-32a^{2}-28a+20$ (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: | \( 6125748579000 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{11}\cdot 6125748579000 \cdot 1}{2\cdot\sqrt{24681102168332647528758926939274761787816079}}\cr\approx \mathstrut & 0.742942914206480 \end{aligned}\] (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 | ${\href{/padicField/2.11.0.1}{11} }^{2}{,}\,{\href{/padicField/2.1.0.1}{1} }$ | ${\href{/padicField/3.10.0.1}{10} }^{2}{,}\,{\href{/padicField/3.2.0.1}{2} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/5.5.0.1}{5} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.10.0.1}{10} }{,}\,{\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | $17{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.12.0.1}{12} }{,}\,{\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.4.0.1}{4} }$ | $22{,}\,{\href{/padicField/23.1.0.1}{1} }$ | $17{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.12.0.1}{12} }{,}\,{\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ | $18{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | $15{,}\,{\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.10.0.1}{10} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $18{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(114089\) | Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $17$ | $1$ | $17$ | $0$ | $C_{17}$ | $[\ ]^{17}$ | ||
\(350671093\) | $\Q_{350671093}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{350671093}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ | ||
\(18936502114367\) | 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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $7$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | ||
\(32577747177637381\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ |