Normalized defining polynomial
\( x^{11} - 1265 x^{9} - 17457 x^{8} + 94116 x^{7} + 2769844 x^{6} + 11844195 x^{5} - 25677729 x^{4} + \cdots + 479405629 \)
Invariants
Degree: | $11$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[11, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(27869685209056005146671841116784449\) \(\medspace = 11^{20}\cdot 23^{10}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(1353.24\) | 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: | $11^{20/11}23^{10/11}\approx 1353.24254149841$ | ||
Ramified primes: | \(11\), \(23\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Gal(K/\Q) }$: | $11$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is Galois and abelian over $\Q$. | |||
Conductor: | \(2783=11^{2}\cdot 23\) | ||
Dirichlet character group: | $\lbrace$$\chi_{2783}(1,·)$, $\chi_{2783}(1189,·)$, $\chi_{2783}(1904,·)$, $\chi_{2783}(749,·)$, $\chi_{2783}(1200,·)$, $\chi_{2783}(1618,·)$, $\chi_{2783}(2740,·)$, $\chi_{2783}(2674,·)$, $\chi_{2783}(1750,·)$, $\chi_{2783}(1849,·)$, $\chi_{2783}(1277,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{99\!\cdots\!39}a^{10}-\frac{13\!\cdots\!62}{99\!\cdots\!39}a^{9}-\frac{83\!\cdots\!02}{99\!\cdots\!39}a^{8}-\frac{40\!\cdots\!56}{99\!\cdots\!39}a^{7}+\frac{16\!\cdots\!87}{99\!\cdots\!39}a^{6}-\frac{83\!\cdots\!55}{99\!\cdots\!39}a^{5}+\frac{10\!\cdots\!57}{99\!\cdots\!39}a^{4}-\frac{74\!\cdots\!65}{99\!\cdots\!39}a^{3}+\frac{30\!\cdots\!10}{99\!\cdots\!39}a^{2}+\frac{79\!\cdots\!06}{99\!\cdots\!39}a+\frac{14\!\cdots\!04}{99\!\cdots\!39}$
Monogenic: | No | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{11}$, which has order $11$ (assuming GRH)
Unit group
Rank: | $10$ | 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: | $\frac{34\!\cdots\!38}{99\!\cdots\!39}a^{10}+\frac{86\!\cdots\!37}{99\!\cdots\!39}a^{9}-\frac{42\!\cdots\!42}{99\!\cdots\!39}a^{8}-\frac{70\!\cdots\!69}{99\!\cdots\!39}a^{7}+\frac{14\!\cdots\!53}{99\!\cdots\!39}a^{6}+\frac{98\!\cdots\!51}{99\!\cdots\!39}a^{5}+\frac{64\!\cdots\!82}{99\!\cdots\!39}a^{4}+\frac{66\!\cdots\!90}{99\!\cdots\!39}a^{3}-\frac{54\!\cdots\!64}{99\!\cdots\!39}a^{2}-\frac{14\!\cdots\!46}{99\!\cdots\!39}a-\frac{73\!\cdots\!99}{99\!\cdots\!39}$, $\frac{16\!\cdots\!33}{99\!\cdots\!39}a^{10}-\frac{14\!\cdots\!85}{99\!\cdots\!39}a^{9}-\frac{19\!\cdots\!51}{99\!\cdots\!39}a^{8}-\frac{11\!\cdots\!55}{99\!\cdots\!39}a^{7}+\frac{25\!\cdots\!61}{99\!\cdots\!39}a^{6}+\frac{22\!\cdots\!85}{99\!\cdots\!39}a^{5}-\frac{80\!\cdots\!38}{99\!\cdots\!39}a^{4}-\frac{35\!\cdots\!60}{99\!\cdots\!39}a^{3}-\frac{22\!\cdots\!44}{99\!\cdots\!39}a^{2}+\frac{13\!\cdots\!31}{99\!\cdots\!39}a+\frac{86\!\cdots\!16}{99\!\cdots\!39}$, $\frac{13\!\cdots\!36}{99\!\cdots\!39}a^{10}-\frac{10\!\cdots\!99}{99\!\cdots\!39}a^{9}-\frac{16\!\cdots\!34}{99\!\cdots\!39}a^{8}-\frac{11\!\cdots\!11}{99\!\cdots\!39}a^{7}+\frac{20\!\cdots\!09}{99\!\cdots\!39}a^{6}+\frac{21\!\cdots\!94}{99\!\cdots\!39}a^{5}+\frac{33\!\cdots\!57}{99\!\cdots\!39}a^{4}-\frac{34\!\cdots\!64}{99\!\cdots\!39}a^{3}-\frac{30\!\cdots\!36}{99\!\cdots\!39}a^{2}+\frac{13\!\cdots\!68}{99\!\cdots\!39}a+\frac{94\!\cdots\!22}{99\!\cdots\!39}$, $\frac{27\!\cdots\!83}{99\!\cdots\!39}a^{10}-\frac{32\!\cdots\!78}{99\!\cdots\!39}a^{9}-\frac{33\!\cdots\!03}{99\!\cdots\!39}a^{8}-\frac{79\!\cdots\!40}{99\!\cdots\!39}a^{7}+\frac{57\!\cdots\!43}{99\!\cdots\!39}a^{6}+\frac{17\!\cdots\!63}{99\!\cdots\!39}a^{5}-\frac{20\!\cdots\!92}{99\!\cdots\!39}a^{4}-\frac{16\!\cdots\!57}{99\!\cdots\!39}a^{3}+\frac{14\!\cdots\!93}{99\!\cdots\!39}a^{2}-\frac{10\!\cdots\!54}{99\!\cdots\!39}a-\frac{15\!\cdots\!66}{99\!\cdots\!39}$, $\frac{61\!\cdots\!62}{99\!\cdots\!39}a^{10}-\frac{39\!\cdots\!82}{99\!\cdots\!39}a^{9}-\frac{77\!\cdots\!86}{99\!\cdots\!39}a^{8}-\frac{58\!\cdots\!46}{99\!\cdots\!39}a^{7}+\frac{11\!\cdots\!11}{99\!\cdots\!39}a^{6}+\frac{12\!\cdots\!67}{99\!\cdots\!39}a^{5}-\frac{24\!\cdots\!10}{99\!\cdots\!39}a^{4}-\frac{49\!\cdots\!17}{99\!\cdots\!39}a^{3}-\frac{46\!\cdots\!24}{99\!\cdots\!39}a^{2}+\frac{29\!\cdots\!36}{99\!\cdots\!39}a+\frac{19\!\cdots\!49}{99\!\cdots\!39}$, $\frac{18\!\cdots\!38}{99\!\cdots\!39}a^{10}-\frac{54\!\cdots\!42}{99\!\cdots\!39}a^{9}-\frac{22\!\cdots\!70}{99\!\cdots\!39}a^{8}-\frac{25\!\cdots\!95}{99\!\cdots\!39}a^{7}+\frac{24\!\cdots\!44}{99\!\cdots\!39}a^{6}+\frac{43\!\cdots\!92}{99\!\cdots\!39}a^{5}+\frac{87\!\cdots\!45}{99\!\cdots\!39}a^{4}-\frac{73\!\cdots\!82}{99\!\cdots\!39}a^{3}-\frac{15\!\cdots\!68}{99\!\cdots\!39}a^{2}+\frac{38\!\cdots\!67}{99\!\cdots\!39}a+\frac{34\!\cdots\!86}{99\!\cdots\!39}$, $\frac{11\!\cdots\!93}{99\!\cdots\!39}a^{10}-\frac{10\!\cdots\!32}{99\!\cdots\!39}a^{9}-\frac{13\!\cdots\!71}{99\!\cdots\!39}a^{8}-\frac{76\!\cdots\!87}{99\!\cdots\!39}a^{7}+\frac{17\!\cdots\!30}{99\!\cdots\!39}a^{6}+\frac{15\!\cdots\!19}{99\!\cdots\!39}a^{5}-\frac{57\!\cdots\!95}{99\!\cdots\!39}a^{4}-\frac{23\!\cdots\!73}{99\!\cdots\!39}a^{3}-\frac{15\!\cdots\!31}{99\!\cdots\!39}a^{2}+\frac{94\!\cdots\!52}{99\!\cdots\!39}a+\frac{60\!\cdots\!91}{99\!\cdots\!39}$, $\frac{14\!\cdots\!46}{99\!\cdots\!39}a^{10}-\frac{57\!\cdots\!30}{99\!\cdots\!39}a^{9}-\frac{18\!\cdots\!11}{99\!\cdots\!39}a^{8}-\frac{18\!\cdots\!78}{99\!\cdots\!39}a^{7}+\frac{21\!\cdots\!95}{99\!\cdots\!39}a^{6}+\frac{32\!\cdots\!22}{99\!\cdots\!39}a^{5}+\frac{46\!\cdots\!86}{99\!\cdots\!39}a^{4}-\frac{56\!\cdots\!64}{99\!\cdots\!39}a^{3}-\frac{80\!\cdots\!26}{99\!\cdots\!39}a^{2}+\frac{25\!\cdots\!93}{99\!\cdots\!39}a+\frac{18\!\cdots\!77}{99\!\cdots\!39}$, $\frac{45\!\cdots\!78}{99\!\cdots\!39}a^{10}-\frac{21\!\cdots\!37}{99\!\cdots\!39}a^{9}-\frac{57\!\cdots\!67}{99\!\cdots\!39}a^{8}-\frac{76\!\cdots\!14}{99\!\cdots\!39}a^{7}+\frac{40\!\cdots\!13}{99\!\cdots\!39}a^{6}+\frac{11\!\cdots\!28}{99\!\cdots\!39}a^{5}+\frac{51\!\cdots\!11}{99\!\cdots\!39}a^{4}-\frac{27\!\cdots\!70}{99\!\cdots\!39}a^{3}-\frac{35\!\cdots\!55}{99\!\cdots\!39}a^{2}-\frac{41\!\cdots\!08}{99\!\cdots\!39}a-\frac{66\!\cdots\!60}{99\!\cdots\!39}$, $\frac{60\!\cdots\!14}{99\!\cdots\!39}a^{10}-\frac{47\!\cdots\!49}{99\!\cdots\!39}a^{9}-\frac{72\!\cdots\!84}{99\!\cdots\!39}a^{8}-\frac{49\!\cdots\!88}{99\!\cdots\!39}a^{7}+\frac{89\!\cdots\!36}{99\!\cdots\!39}a^{6}+\frac{94\!\cdots\!33}{99\!\cdots\!39}a^{5}+\frac{53\!\cdots\!65}{99\!\cdots\!39}a^{4}-\frac{13\!\cdots\!69}{99\!\cdots\!39}a^{3}-\frac{16\!\cdots\!85}{99\!\cdots\!39}a^{2}+\frac{43\!\cdots\!67}{99\!\cdots\!39}a+\frac{31\!\cdots\!83}{99\!\cdots\!39}$ (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: | \( 2107938978012.7551 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{0}\cdot 2107938978012.7551 \cdot 11}{2\cdot\sqrt{27869685209056005146671841116784449}}\cr\approx \mathstrut & 0.142227849860664 \end{aligned}\] (assuming GRH)
Galois group
A cyclic group of order 11 |
The 11 conjugacy class representatives for $C_{11}$ |
Character table for $C_{11}$ |
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 | ${\href{/padicField/2.11.0.1}{11} }$ | ${\href{/padicField/3.11.0.1}{11} }$ | ${\href{/padicField/5.11.0.1}{11} }$ | ${\href{/padicField/7.11.0.1}{11} }$ | R | ${\href{/padicField/13.11.0.1}{11} }$ | ${\href{/padicField/17.11.0.1}{11} }$ | ${\href{/padicField/19.11.0.1}{11} }$ | R | ${\href{/padicField/29.11.0.1}{11} }$ | ${\href{/padicField/31.11.0.1}{11} }$ | ${\href{/padicField/37.11.0.1}{11} }$ | ${\href{/padicField/41.11.0.1}{11} }$ | ${\href{/padicField/43.11.0.1}{11} }$ | ${\href{/padicField/47.11.0.1}{11} }$ | ${\href{/padicField/53.11.0.1}{11} }$ | ${\href{/padicField/59.11.0.1}{11} }$ |
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 |
---|---|---|---|---|---|---|---|
\(11\) | 11.11.20.8 | $x^{11} + 110 x^{10} + 1221$ | $11$ | $1$ | $20$ | $C_{11}$ | $[2]$ |
\(23\) | 23.11.10.11 | $x^{11} + 92$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ |