Normalized defining polynomial
\( x^{11} - 1265 x^{9} - 759 x^{8} + 461472 x^{7} + 727122 x^{6} - 64557504 x^{5} - 152866395 x^{4} + 3259832136 x^{3} + 7537075436 x^{2} + \cdots - 79493912081 \)
Invariants
Degree: | $11$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
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)
| |
This field is Galois and abelian over $\Q$. | |||
Conductor: | \(2783=11^{2}\cdot 23\) | ||
Dirichlet character group: | $\lbrace$$\chi_{2783}(1,·)$, $\chi_{2783}(1475,·)$, $\chi_{2783}(100,·)$, $\chi_{2783}(903,·)$, $\chi_{2783}(1244,·)$, $\chi_{2783}(1948,·)$, $\chi_{2783}(1651,·)$, $\chi_{2783}(2773,·)$, $\chi_{2783}(2102,·)$, $\chi_{2783}(1783,·)$, $\chi_{2783}(188,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{42\!\cdots\!97}a^{10}-\frac{65\!\cdots\!33}{42\!\cdots\!97}a^{9}-\frac{17\!\cdots\!94}{42\!\cdots\!97}a^{8}+\frac{70\!\cdots\!96}{42\!\cdots\!97}a^{7}+\frac{93\!\cdots\!88}{42\!\cdots\!97}a^{6}-\frac{17\!\cdots\!20}{42\!\cdots\!97}a^{5}+\frac{82\!\cdots\!38}{42\!\cdots\!97}a^{4}+\frac{15\!\cdots\!90}{42\!\cdots\!97}a^{3}-\frac{83\!\cdots\!70}{42\!\cdots\!97}a^{2}-\frac{18\!\cdots\!71}{42\!\cdots\!97}a-\frac{48\!\cdots\!40}{42\!\cdots\!97}$
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: | $\frac{39\!\cdots\!23}{42\!\cdots\!97}a^{10}-\frac{40\!\cdots\!96}{42\!\cdots\!97}a^{9}-\frac{45\!\cdots\!26}{42\!\cdots\!97}a^{8}+\frac{43\!\cdots\!01}{42\!\cdots\!97}a^{7}+\frac{14\!\cdots\!97}{42\!\cdots\!97}a^{6}-\frac{11\!\cdots\!37}{42\!\cdots\!97}a^{5}-\frac{16\!\cdots\!56}{42\!\cdots\!97}a^{4}+\frac{95\!\cdots\!08}{42\!\cdots\!97}a^{3}+\frac{60\!\cdots\!56}{42\!\cdots\!97}a^{2}-\frac{22\!\cdots\!39}{42\!\cdots\!97}a-\frac{41\!\cdots\!48}{42\!\cdots\!97}$, $\frac{90\!\cdots\!98}{42\!\cdots\!97}a^{10}-\frac{64\!\cdots\!53}{42\!\cdots\!97}a^{9}-\frac{10\!\cdots\!78}{42\!\cdots\!97}a^{8}+\frac{71\!\cdots\!78}{42\!\cdots\!97}a^{7}+\frac{36\!\cdots\!09}{42\!\cdots\!97}a^{6}-\frac{19\!\cdots\!05}{42\!\cdots\!97}a^{5}-\frac{44\!\cdots\!61}{42\!\cdots\!97}a^{4}+\frac{17\!\cdots\!50}{42\!\cdots\!97}a^{3}+\frac{16\!\cdots\!86}{42\!\cdots\!97}a^{2}-\frac{50\!\cdots\!79}{42\!\cdots\!97}a-\frac{99\!\cdots\!80}{42\!\cdots\!97}$, $\frac{21\!\cdots\!53}{42\!\cdots\!97}a^{10}+\frac{20\!\cdots\!55}{42\!\cdots\!97}a^{9}-\frac{21\!\cdots\!62}{42\!\cdots\!97}a^{8}-\frac{21\!\cdots\!93}{42\!\cdots\!97}a^{7}+\frac{42\!\cdots\!38}{42\!\cdots\!97}a^{6}+\frac{48\!\cdots\!20}{42\!\cdots\!97}a^{5}-\frac{16\!\cdots\!14}{42\!\cdots\!97}a^{4}-\frac{28\!\cdots\!67}{42\!\cdots\!97}a^{3}-\frac{84\!\cdots\!12}{42\!\cdots\!97}a^{2}+\frac{43\!\cdots\!68}{42\!\cdots\!97}a+\frac{57\!\cdots\!74}{42\!\cdots\!97}$, $\frac{27\!\cdots\!21}{42\!\cdots\!97}a^{10}+\frac{61\!\cdots\!93}{42\!\cdots\!97}a^{9}+\frac{39\!\cdots\!49}{42\!\cdots\!97}a^{8}-\frac{67\!\cdots\!03}{42\!\cdots\!97}a^{7}-\frac{72\!\cdots\!33}{42\!\cdots\!97}a^{6}+\frac{16\!\cdots\!07}{42\!\cdots\!97}a^{5}+\frac{20\!\cdots\!79}{42\!\cdots\!97}a^{4}-\frac{91\!\cdots\!85}{42\!\cdots\!97}a^{3}-\frac{14\!\cdots\!97}{42\!\cdots\!97}a^{2}+\frac{90\!\cdots\!96}{42\!\cdots\!97}a+\frac{24\!\cdots\!62}{42\!\cdots\!97}$, $\frac{89\!\cdots\!81}{42\!\cdots\!97}a^{10}-\frac{65\!\cdots\!82}{42\!\cdots\!97}a^{9}-\frac{10\!\cdots\!85}{42\!\cdots\!97}a^{8}+\frac{73\!\cdots\!69}{42\!\cdots\!97}a^{7}+\frac{35\!\cdots\!49}{42\!\cdots\!97}a^{6}-\frac{20\!\cdots\!58}{42\!\cdots\!97}a^{5}-\frac{43\!\cdots\!68}{42\!\cdots\!97}a^{4}+\frac{18\!\cdots\!75}{42\!\cdots\!97}a^{3}+\frac{15\!\cdots\!32}{42\!\cdots\!97}a^{2}-\frac{50\!\cdots\!69}{42\!\cdots\!97}a-\frac{97\!\cdots\!34}{42\!\cdots\!97}$, $\frac{23\!\cdots\!18}{42\!\cdots\!97}a^{10}+\frac{32\!\cdots\!08}{42\!\cdots\!97}a^{9}-\frac{35\!\cdots\!98}{42\!\cdots\!97}a^{8}-\frac{39\!\cdots\!08}{42\!\cdots\!97}a^{7}+\frac{16\!\cdots\!28}{42\!\cdots\!97}a^{6}+\frac{13\!\cdots\!66}{42\!\cdots\!97}a^{5}-\frac{16\!\cdots\!67}{42\!\cdots\!97}a^{4}-\frac{16\!\cdots\!71}{42\!\cdots\!97}a^{3}-\frac{10\!\cdots\!39}{42\!\cdots\!97}a^{2}+\frac{53\!\cdots\!93}{42\!\cdots\!97}a+\frac{74\!\cdots\!72}{42\!\cdots\!97}$, $\frac{12\!\cdots\!38}{42\!\cdots\!97}a^{10}-\frac{93\!\cdots\!45}{42\!\cdots\!97}a^{9}-\frac{15\!\cdots\!84}{42\!\cdots\!97}a^{8}+\frac{10\!\cdots\!07}{42\!\cdots\!97}a^{7}+\frac{51\!\cdots\!37}{42\!\cdots\!97}a^{6}-\frac{28\!\cdots\!70}{42\!\cdots\!97}a^{5}-\frac{61\!\cdots\!50}{42\!\cdots\!97}a^{4}+\frac{25\!\cdots\!97}{42\!\cdots\!97}a^{3}+\frac{22\!\cdots\!11}{42\!\cdots\!97}a^{2}-\frac{71\!\cdots\!07}{42\!\cdots\!97}a-\frac{13\!\cdots\!32}{42\!\cdots\!97}$, $\frac{42\!\cdots\!06}{42\!\cdots\!97}a^{10}-\frac{67\!\cdots\!64}{42\!\cdots\!97}a^{9}-\frac{46\!\cdots\!50}{42\!\cdots\!97}a^{8}+\frac{69\!\cdots\!42}{42\!\cdots\!97}a^{7}+\frac{12\!\cdots\!05}{42\!\cdots\!97}a^{6}-\frac{15\!\cdots\!15}{42\!\cdots\!97}a^{5}-\frac{13\!\cdots\!24}{42\!\cdots\!97}a^{4}+\frac{98\!\cdots\!97}{42\!\cdots\!97}a^{3}+\frac{48\!\cdots\!93}{42\!\cdots\!97}a^{2}-\frac{19\!\cdots\!52}{42\!\cdots\!97}a-\frac{35\!\cdots\!24}{42\!\cdots\!97}$, $\frac{73\!\cdots\!51}{42\!\cdots\!97}a^{10}-\frac{52\!\cdots\!00}{42\!\cdots\!97}a^{9}-\frac{89\!\cdots\!21}{42\!\cdots\!97}a^{8}+\frac{58\!\cdots\!73}{42\!\cdots\!97}a^{7}+\frac{29\!\cdots\!45}{42\!\cdots\!97}a^{6}-\frac{16\!\cdots\!34}{42\!\cdots\!97}a^{5}-\frac{36\!\cdots\!39}{42\!\cdots\!97}a^{4}+\frac{14\!\cdots\!07}{42\!\cdots\!97}a^{3}+\frac{13\!\cdots\!02}{42\!\cdots\!97}a^{2}-\frac{41\!\cdots\!51}{42\!\cdots\!97}a-\frac{81\!\cdots\!29}{42\!\cdots\!97}$, $\frac{19\!\cdots\!14}{42\!\cdots\!97}a^{10}+\frac{15\!\cdots\!90}{42\!\cdots\!97}a^{9}-\frac{24\!\cdots\!35}{42\!\cdots\!97}a^{8}-\frac{20\!\cdots\!46}{42\!\cdots\!97}a^{7}+\frac{78\!\cdots\!25}{42\!\cdots\!97}a^{6}+\frac{75\!\cdots\!52}{42\!\cdots\!97}a^{5}-\frac{77\!\cdots\!29}{42\!\cdots\!97}a^{4}-\frac{92\!\cdots\!34}{42\!\cdots\!97}a^{3}+\frac{26\!\cdots\!23}{42\!\cdots\!97}a^{2}+\frac{19\!\cdots\!66}{42\!\cdots\!97}a+\frac{24\!\cdots\!78}{42\!\cdots\!97}$ (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: | \( 2768761764338.989 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{0}\cdot 2768761764338.989 \cdot 11}{2\cdot\sqrt{27869685209056005146671841116784449}}\cr\approx \mathstrut & 0.186815195613300 \end{aligned}\] (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 | ${\href{/padicField/29.11.0.1}{11} }$ | ${\href{/padicField/31.11.0.1}{11} }$ | ${\href{/padicField/37.1.0.1}{1} }^{11}$ | ${\href{/padicField/41.11.0.1}{11} }$ | ${\href{/padicField/43.11.0.1}{11} }$ | ${\href{/padicField/47.11.0.1}{11} }$ | ${\href{/padicField/53.11.0.1}{11} }$ | ${\href{/padicField/59.11.0.1}{11} }$ |
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 |
---|---|---|---|---|---|---|---|
\(11\) | 11.11.20.11 | $x^{11} + 110 x^{10} + 253$ | $11$ | $1$ | $20$ | $C_{11}$ | $[2]$ |
\(23\) | 23.11.10.8 | $x^{11} + 253$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ |