Normalized defining polynomial
\( x^{11} + 15x^{9} - 2x^{8} + 90x^{7} + 24x^{6} - 64x^{5} + 300x^{4} - 1203x^{3} + 240x^{2} + 45x + 54 \)
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: | \(4740548198400000000\) \(\medspace = 2^{22}\cdot 3^{10}\cdot 5^{8}\cdot 7^{2}\) | 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: | \(49.87\) | 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: | $2^{35/12}3^{79/54}5^{23/20}7^{2/3}\approx 877.4635338525777$ | ||
Ramified primes: | \(2\), \(3\), \(5\), \(7\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{30}a^{9}-\frac{1}{10}a^{8}+\frac{1}{10}a^{7}+\frac{7}{30}a^{6}-\frac{3}{10}a^{5}+\frac{3}{10}a^{4}-\frac{7}{30}a^{3}-\frac{1}{10}a^{2}-\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{198856615950}a^{10}+\frac{605284657}{66285538650}a^{9}-\frac{3569837999}{33142769325}a^{8}-\frac{13960278388}{99428307975}a^{7}+\frac{12334360499}{33142769325}a^{6}-\frac{12410926667}{33142769325}a^{5}-\frac{34854185678}{99428307975}a^{4}-\frac{524291032}{11047589775}a^{3}-\frac{29494948433}{66285538650}a^{2}-\frac{13641065413}{66285538650}a+\frac{3044689907}{11047589775}$
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: | $\frac{616378957}{198856615950}a^{10}+\frac{450103909}{66285538650}a^{9}+\frac{3478127159}{66285538650}a^{8}+\frac{15208151683}{198856615950}a^{7}+\frac{18436771831}{66285538650}a^{6}+\frac{25271683997}{66285538650}a^{5}+\frac{8373706103}{198856615950}a^{4}+\frac{16991285437}{22095179550}a^{3}-\frac{57434026018}{33142769325}a^{2}-\frac{23974261868}{33142769325}a+\frac{8128565639}{11047589775}$, $\frac{417552187}{19885661595}a^{10}-\frac{2228507}{6628553865}a^{9}+\frac{2114370347}{6628553865}a^{8}-\frac{433909601}{19885661595}a^{7}+\frac{12743593264}{6628553865}a^{6}+\frac{4767825896}{6628553865}a^{5}-\frac{27300400879}{19885661595}a^{4}+\frac{16440065101}{2209517955}a^{3}-\frac{172925754158}{6628553865}a^{2}-\frac{5067096274}{6628553865}a+\frac{2756603393}{441903591}$, $\frac{1011835496}{99428307975}a^{10}+\frac{15626809}{66285538650}a^{9}+\frac{9396785489}{66285538650}a^{8}+\frac{1376467093}{198856615950}a^{7}+\frac{52602241771}{66285538650}a^{6}+\frac{23111888237}{66285538650}a^{5}-\frac{149874157147}{198856615950}a^{4}+\frac{55025617327}{22095179550}a^{3}-\frac{661366310681}{66285538650}a^{2}-\frac{4771627988}{33142769325}a+\frac{26555196629}{11047589775}$, $\frac{708099337}{99428307975}a^{10}+\frac{29861213}{66285538650}a^{9}+\frac{1852091819}{33142769325}a^{8}-\frac{11059557347}{99428307975}a^{7}+\frac{1155191596}{33142769325}a^{6}-\frac{4424856298}{33142769325}a^{5}-\frac{117090599302}{99428307975}a^{4}+\frac{22116441892}{11047589775}a^{3}-\frac{1380833951}{33142769325}a^{2}+\frac{3156002473}{66285538650}a-\frac{574534052}{11047589775}$, $\frac{145121183}{19885661595}a^{10}+\frac{20944300}{1325710773}a^{9}+\frac{1290102833}{13257107730}a^{8}+\frac{5481700141}{39771323190}a^{7}+\frac{10263176149}{13257107730}a^{6}+\frac{5388062819}{13257107730}a^{5}+\frac{1987443643}{7954264638}a^{4}+\frac{4080895129}{4419035910}a^{3}-\frac{126122950007}{13257107730}a^{2}-\frac{36749112169}{13257107730}a-\frac{2071000396}{2209517955}$, $\frac{187079426189}{198856615950}a^{10}+\frac{95289413033}{66285538650}a^{9}+\frac{540515933249}{33142769325}a^{8}+\frac{2268078593113}{99428307975}a^{7}+\frac{3948310229896}{33142769325}a^{6}+\frac{6682171374692}{33142769325}a^{5}+\frac{24229974457193}{99428307975}a^{4}+\frac{7057870143457}{11047589775}a^{3}-\frac{12473720696767}{66285538650}a^{2}-\frac{3434874210227}{66285538650}a-\frac{379581240812}{11047589775}$ (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)]
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Regulator: | \( 2661549.44457 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{4}\cdot 2661549.44457 \cdot 1}{2\cdot\sqrt{4740548198400000000}}\cr\approx \mathstrut & 7.62078518923 \end{aligned}\] (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 | R | R | ${\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.11.0.1}{11} }$ | ${\href{/padicField/17.11.0.1}{11} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.11.0.1}{11} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.11.0.1}{11} }$ |
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 | $[\ ]$ |
2.4.11.13 | $x^{4} + 8 x^{3} + 8 x + 10$ | $4$ | $1$ | $11$ | $D_{4}$ | $[3, 4]^{2}$ | |
2.6.11.9 | $x^{6} + 4 x^{5} + 4 x^{2} + 2$ | $6$ | $1$ | $11$ | $D_{6}$ | $[3]_{3}^{2}$ | |
\(3\) | 3.2.1.1 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
3.3.3.1 | $x^{3} + 6 x + 3$ | $3$ | $1$ | $3$ | $S_3$ | $[3/2]_{2}$ | |
3.6.6.1 | $x^{6} - 6 x^{5} + 24 x^{4} + 6 x^{3} + 18 x + 9$ | $3$ | $2$ | $6$ | $C_3^2:C_4$ | $[3/2, 3/2]_{2}^{2}$ | |
\(5\) | 5.2.1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
5.4.2.1 | $x^{4} + 48 x^{3} + 670 x^{2} + 2256 x + 1449$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
5.5.5.4 | $x^{5} + 10 x + 5$ | $5$ | $1$ | $5$ | $F_5$ | $[5/4]_{4}$ | |
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.3.2.2 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $[\ ]_{3}$ | |
7.7.0.1 | $x^{7} + 6 x + 4$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ |