Normalized defining polynomial
\( x^{11} - 66 x^{9} - 484 x^{8} + 5643 x^{7} + 924 x^{6} + 207328 x^{5} - 1002936 x^{4} - 1198032 x^{3} + \cdots + 9899136 \)
Invariants
Degree: | $11$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[3, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(8296271242435216551954427674624\) \(\medspace = 2^{22}\cdot 3^{16}\cdot 11^{16}\) | 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: | \(646.86\) | 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))
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Galois root discriminant: | $2^{25/8}3^{37/18}11^{84/55}\approx 3250.5633072887913$ | ||
Ramified primes: | \(2\), \(3\), \(11\) | 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: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{64}a^{7}-\frac{1}{8}a^{6}+\frac{3}{32}a^{5}+\frac{3}{16}a^{4}+\frac{11}{64}a^{3}-\frac{7}{16}a^{2}-\frac{1}{8}a+\frac{1}{8}$, $\frac{1}{1024}a^{8}-\frac{1}{512}a^{7}+\frac{11}{512}a^{6}+\frac{11}{64}a^{5}+\frac{275}{1024}a^{4}-\frac{237}{512}a^{3}-\frac{11}{64}a^{2}+\frac{27}{128}a-\frac{29}{64}$, $\frac{1}{16384}a^{9}+\frac{1}{4096}a^{8}+\frac{5}{8192}a^{7}-\frac{51}{4096}a^{6}-\frac{2765}{16384}a^{5}-\frac{749}{2048}a^{4}+\frac{269}{4096}a^{3}+\frac{215}{2048}a^{2}-\frac{67}{256}a+\frac{169}{512}$, $\frac{1}{16\!\cdots\!84}a^{10}-\frac{27\!\cdots\!77}{26\!\cdots\!64}a^{9}+\frac{11\!\cdots\!15}{26\!\cdots\!64}a^{8}+\frac{83\!\cdots\!83}{10\!\cdots\!24}a^{7}+\frac{41\!\cdots\!57}{53\!\cdots\!28}a^{6}+\frac{90\!\cdots\!79}{26\!\cdots\!64}a^{5}-\frac{66\!\cdots\!39}{40\!\cdots\!96}a^{4}-\frac{51\!\cdots\!25}{11\!\cdots\!36}a^{3}+\frac{10\!\cdots\!23}{33\!\cdots\!08}a^{2}-\frac{34\!\cdots\!87}{50\!\cdots\!12}a-\frac{25\!\cdots\!19}{83\!\cdots\!52}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $6$ | 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)
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Fundamental units: | $\frac{90\!\cdots\!41}{80\!\cdots\!92}a^{10}-\frac{23\!\cdots\!05}{13\!\cdots\!32}a^{9}+\frac{13\!\cdots\!07}{13\!\cdots\!32}a^{8}-\frac{22\!\cdots\!63}{50\!\cdots\!12}a^{7}+\frac{82\!\cdots\!49}{26\!\cdots\!64}a^{6}+\frac{10\!\cdots\!83}{13\!\cdots\!32}a^{5}-\frac{18\!\cdots\!95}{20\!\cdots\!48}a^{4}+\frac{24\!\cdots\!47}{55\!\cdots\!68}a^{3}-\frac{17\!\cdots\!93}{16\!\cdots\!04}a^{2}-\frac{49\!\cdots\!91}{25\!\cdots\!56}a+\frac{31\!\cdots\!73}{41\!\cdots\!76}$, $\frac{29\!\cdots\!47}{13\!\cdots\!32}a^{10}-\frac{22\!\cdots\!19}{22\!\cdots\!72}a^{9}-\frac{53\!\cdots\!59}{22\!\cdots\!72}a^{8}-\frac{51\!\cdots\!49}{83\!\cdots\!52}a^{7}+\frac{97\!\cdots\!23}{44\!\cdots\!44}a^{6}-\frac{25\!\cdots\!27}{22\!\cdots\!72}a^{5}-\frac{15\!\cdots\!25}{33\!\cdots\!08}a^{4}+\frac{13\!\cdots\!09}{93\!\cdots\!28}a^{3}+\frac{13\!\cdots\!45}{27\!\cdots\!84}a^{2}-\frac{25\!\cdots\!17}{41\!\cdots\!76}a+\frac{11\!\cdots\!55}{69\!\cdots\!96}$, $\frac{38\!\cdots\!65}{80\!\cdots\!92}a^{10}-\frac{30\!\cdots\!29}{13\!\cdots\!32}a^{9}+\frac{44\!\cdots\!39}{13\!\cdots\!32}a^{8}+\frac{10\!\cdots\!87}{50\!\cdots\!12}a^{7}+\frac{10\!\cdots\!57}{26\!\cdots\!64}a^{6}-\frac{30\!\cdots\!57}{13\!\cdots\!32}a^{5}+\frac{14\!\cdots\!17}{20\!\cdots\!48}a^{4}+\frac{12\!\cdots\!23}{55\!\cdots\!68}a^{3}-\frac{34\!\cdots\!77}{16\!\cdots\!04}a^{2}+\frac{48\!\cdots\!65}{25\!\cdots\!56}a-\frac{19\!\cdots\!43}{41\!\cdots\!76}$, $\frac{37\!\cdots\!85}{26\!\cdots\!64}a^{10}-\frac{11\!\cdots\!53}{44\!\cdots\!44}a^{9}-\frac{27\!\cdots\!29}{44\!\cdots\!44}a^{8}-\frac{93\!\cdots\!37}{16\!\cdots\!04}a^{7}-\frac{12\!\cdots\!03}{89\!\cdots\!88}a^{6}-\frac{29\!\cdots\!81}{44\!\cdots\!44}a^{5}+\frac{13\!\cdots\!33}{67\!\cdots\!16}a^{4}-\frac{59\!\cdots\!93}{18\!\cdots\!56}a^{3}-\frac{11\!\cdots\!81}{55\!\cdots\!68}a^{2}+\frac{19\!\cdots\!85}{83\!\cdots\!52}a-\frac{88\!\cdots\!47}{13\!\cdots\!92}$, $\frac{77\!\cdots\!03}{20\!\cdots\!48}a^{10}-\frac{37\!\cdots\!81}{33\!\cdots\!08}a^{9}-\frac{78\!\cdots\!99}{33\!\cdots\!08}a^{8}-\frac{51\!\cdots\!91}{25\!\cdots\!56}a^{7}-\frac{33\!\cdots\!33}{67\!\cdots\!16}a^{6}-\frac{78\!\cdots\!85}{33\!\cdots\!08}a^{5}+\frac{30\!\cdots\!51}{50\!\cdots\!12}a^{4}-\frac{41\!\cdots\!25}{34\!\cdots\!48}a^{3}-\frac{30\!\cdots\!89}{41\!\cdots\!76}a^{2}+\frac{52\!\cdots\!23}{62\!\cdots\!64}a-\frac{23\!\cdots\!31}{10\!\cdots\!44}$, $\frac{10\!\cdots\!85}{80\!\cdots\!92}a^{10}+\frac{14\!\cdots\!91}{13\!\cdots\!32}a^{9}-\frac{10\!\cdots\!57}{13\!\cdots\!32}a^{8}-\frac{30\!\cdots\!67}{50\!\cdots\!12}a^{7}+\frac{17\!\cdots\!93}{26\!\cdots\!64}a^{6}+\frac{74\!\cdots\!67}{13\!\cdots\!32}a^{5}+\frac{49\!\cdots\!01}{20\!\cdots\!48}a^{4}-\frac{59\!\cdots\!85}{55\!\cdots\!68}a^{3}-\frac{31\!\cdots\!53}{16\!\cdots\!04}a^{2}+\frac{11\!\cdots\!53}{25\!\cdots\!56}a-\frac{90\!\cdots\!23}{41\!\cdots\!76}$ (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: | \( 3626466019480 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{4}\cdot 3626466019480 \cdot 1}{2\cdot\sqrt{8296271242435216551954427674624}}\cr\approx \mathstrut & 7.84913151398559 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 19958400 |
The 31 conjugacy class representatives for $A_{11}$ |
Character table for $A_{11}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
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 | ${\href{/padicField/5.5.0.1}{5} }^{2}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | R | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.11.0.1}{11} }$ | ${\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.9.0.1}{9} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.11.0.1}{11} }$ | ${\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.11.0.1}{11} }$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.2.0.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
2.4.11.11 | $x^{4} + 10$ | $4$ | $1$ | $11$ | $D_{4}$ | $[2, 3, 4]$ | |
2.4.11.6 | $x^{4} + 18$ | $4$ | $1$ | $11$ | $D_{4}$ | $[2, 3, 4]$ | |
\(3\) | 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.3.5.3 | $x^{3} + 9 x + 3$ | $3$ | $1$ | $5$ | $S_3$ | $[5/2]_{2}$ | |
3.6.10.3 | $x^{6} + 6 x^{5} + 36 x^{4} + 128 x^{3} + 297 x^{2} + 474 x + 482$ | $3$ | $2$ | $10$ | $D_{6}$ | $[5/2]_{2}^{2}$ | |
\(11\) | 11.11.16.4 | $x^{11} + 22 x^{6} + 11$ | $11$ | $1$ | $16$ | $C_{11}:C_5$ | $[8/5]_{5}$ |