Normalized defining polynomial
\( x^{23} - 2 x^{22} - 4 x^{21} + 6 x^{20} + 10 x^{19} - 7 x^{18} - 14 x^{17} + 11 x^{16} + 12 x^{15} - 31 x^{14} - 36 x^{13} + 41 x^{12} + 89 x^{11} + 5 x^{10} - 105 x^{9} - 65 x^{8} + 56 x^{7} + 68 x^{6} + \cdots - 1 \)
Invariants
Degree: | $23$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[9, 7]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-30040962001154759467875853831400403771043\) \(\medspace = -\,508621\cdot 158113511\cdot 1092887040821\cdot 341802543194293\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(57.53\) | 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: | $508621^{1/2}158113511^{1/2}1092887040821^{1/2}341802543194293^{1/2}\approx 1.7332328753273392e+20$ | ||
Ramified primes: | \(508621\), \(158113511\), \(1092887040821\), \(341802543194293\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
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Discriminant root field: | $\Q(\sqrt{-30040\!\cdots\!71043}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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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: | $15$ | 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: | $a$, $a^{21}-2a^{20}-3a^{19}+4a^{18}+7a^{17}-3a^{16}-7a^{15}+8a^{14}+5a^{13}-23a^{12}-31a^{11}+18a^{10}+58a^{9}+23a^{8}-47a^{7}-42a^{6}+9a^{5}+26a^{4}+2a^{3}-8a^{2}-2a+1$, $a^{22}-2a^{21}-4a^{20}+6a^{19}+10a^{18}-7a^{17}-14a^{16}+11a^{15}+12a^{14}-31a^{13}-36a^{12}+41a^{11}+89a^{10}+5a^{9}-105a^{8}-65a^{7}+56a^{6}+68a^{5}-7a^{4}-34a^{3}-4a^{2}+9a$, $a^{22}-a^{21}-6a^{20}+2a^{19}+16a^{18}+3a^{17}-21a^{16}-3a^{15}+23a^{14}-19a^{13}-67a^{12}+5a^{11}+130a^{10}+94a^{9}-100a^{8}-170a^{7}-9a^{6}+124a^{5}+61a^{4}-41a^{3}-38a^{2}+4a+9$, $3a^{22}+a^{21}-26a^{20}-9a^{19}+70a^{18}+45a^{17}-85a^{16}-55a^{15}+106a^{14}-23a^{13}-314a^{12}-117a^{11}+523a^{10}+602a^{9}-239a^{8}-841a^{7}-282a^{6}+491a^{5}+390a^{4}-95a^{3}-182a^{2}-8a+33$, $33a^{22}-51a^{21}-188a^{20}+178a^{19}+523a^{18}-150a^{17}-793a^{16}+134a^{15}+762a^{14}-950a^{13}-1833a^{12}+1366a^{11}+4619a^{10}+1437a^{9}-5120a^{8}-5174a^{7}+1564a^{6}+4735a^{5}+1369a^{4}-1701a^{3}-1066a^{2}+199a+184$, $79a^{22}-125a^{21}-347a^{20}+288a^{19}+833a^{18}-103a^{17}-966a^{16}+378a^{15}+888a^{14}-1901a^{13}-3472a^{12}+1227a^{11}+6816a^{10}+3799a^{9}-5123a^{8}-6801a^{7}+128a^{6}+4173a^{5}+1615a^{4}-1122a^{3}-678a^{2}+124a+73$, $8a^{22}-10a^{21}-43a^{20}+23a^{19}+110a^{18}+6a^{17}-138a^{16}+7a^{15}+141a^{14}-179a^{13}-451a^{12}+93a^{11}+891a^{10}+579a^{9}-680a^{8}-1056a^{7}-42a^{6}+710a^{5}+342a^{4}-191a^{3}-173a^{2}+13a+27$, $156a^{22}-227a^{21}-747a^{20}+529a^{19}+1844a^{18}-90a^{17}-2225a^{16}+515a^{15}+2150a^{14}-3672a^{13}-7610a^{12}+2244a^{11}+15057a^{10}+8934a^{9}-11469a^{8}-16268a^{7}-68a^{6}+10501a^{5}+4527a^{4}-2856a^{3}-2139a^{2}+266a+298$, $70a^{22}-106a^{21}-323a^{20}+246a^{19}+790a^{18}-63a^{17}-941a^{16}+272a^{15}+887a^{14}-1660a^{13}-3265a^{12}+1055a^{11}+6472a^{10}+3739a^{9}-4920a^{8}-6808a^{7}+9a^{6}+4308a^{5}+1821a^{4}-1147a^{3}-819a^{2}+103a+106$, $191a^{22}-271a^{21}-911a^{20}+597a^{19}+2218a^{18}-a^{17}-2585a^{16}+566a^{15}+2518a^{14}-4381a^{13}-9341a^{12}+2136a^{11}+17869a^{10}+11565a^{9}-12557a^{8}-19410a^{7}-1229a^{6}+11637a^{5}+5550a^{4}-2867a^{3}-2350a^{2}+236a+300$, $64a^{22}-125a^{21}-241a^{20}+330a^{19}+582a^{18}-309a^{17}-734a^{16}+560a^{15}+571a^{14}-1762a^{13}-2225a^{12}+1940a^{11}+5115a^{10}+1225a^{9}-5078a^{8}-4102a^{7}+1778a^{6}+3251a^{5}+328a^{4}-1198a^{3}-275a^{2}+193a+18$, $58a^{22}-38a^{21}-298a^{20}-72a^{19}+609a^{18}+543a^{17}-340a^{16}-214a^{15}+529a^{14}-767a^{13}-3416a^{12}-2242a^{11}+3692a^{10}+6959a^{9}+1969a^{8}-4989a^{7}-5219a^{6}-449a^{5}+2217a^{4}+1247a^{3}-43a^{2}-239a-64$, $122a^{22}-165a^{21}-621a^{20}+392a^{19}+1543a^{18}-8a^{17}-1888a^{16}+297a^{15}+1881a^{14}-2854a^{13}-6348a^{12}+1607a^{11}+12534a^{10}+7740a^{9}-9481a^{8}-14024a^{7}-265a^{6}+9125a^{5}+4100a^{4}-2489a^{3}-1976a^{2}+230a+281$, $2a^{22}+32a^{21}-37a^{20}-177a^{19}+16a^{18}+400a^{17}+225a^{16}-306a^{15}-67a^{14}+355a^{13}-642a^{12}-2026a^{11}-717a^{10}+2804a^{9}+3669a^{8}-95a^{7}-3478a^{6}-2281a^{5}+664a^{4}+1357a^{3}+319a^{2}-210a-86$ (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: | \( 394956948058 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{7}\cdot 394956948058 \cdot 1}{2\cdot\sqrt{30040962001154759467875853831400403771043}}\cr\approx \mathstrut & 0.225523542178391 \end{aligned}\] (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 | ${\href{/padicField/2.13.0.1}{13} }{,}\,{\href{/padicField/2.10.0.1}{10} }$ | ${\href{/padicField/3.13.0.1}{13} }{,}\,{\href{/padicField/3.10.0.1}{10} }$ | $21{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.9.0.1}{9} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | $18{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.13.0.1}{13} }{,}\,{\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | $16{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.12.0.1}{12} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | $21{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.11.0.1}{11} }{,}\,{\href{/padicField/37.9.0.1}{9} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ | $23$ | $16{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.9.0.1}{9} }{,}\,{\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.3.0.1}{3} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | $22{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(508621\) | $\Q_{508621}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $14$ | $1$ | $14$ | $0$ | $C_{14}$ | $[\ ]^{14}$ | ||
\(158113511\) | $\Q_{158113511}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{158113511}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{158113511}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ | ||
Deg $10$ | $1$ | $10$ | $0$ | $C_{10}$ | $[\ ]^{10}$ | ||
\(1092887040821\) | 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 $16$ | $1$ | $16$ | $0$ | $C_{16}$ | $[\ ]^{16}$ | ||
\(341802543194293\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $19$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ |