Normalized defining polynomial
\( x^{11} - 55 x^{9} - 33 x^{8} + 825 x^{7} + 396 x^{6} - 4972 x^{5} - 1287 x^{4} + 12760 x^{3} + \cdots - 243 \)
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: | \(672749994932560009201\) \(\medspace = 11^{20}\) | 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: | \(78.24\) | 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))
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Galois root discriminant: | $11^{20/11}\approx 78.24223465860136$ | ||
Ramified primes: | \(11\) | 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: | \(121=11^{2}\) | ||
Dirichlet character group: | $\lbrace$$\chi_{121}(1,·)$, $\chi_{121}(34,·)$, $\chi_{121}(67,·)$, $\chi_{121}(100,·)$, $\chi_{121}(12,·)$, $\chi_{121}(45,·)$, $\chi_{121}(78,·)$, $\chi_{121}(111,·)$, $\chi_{121}(23,·)$, $\chi_{121}(56,·)$, $\chi_{121}(89,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}-\frac{1}{3}a$, $\frac{1}{3}a^{4}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{5}-\frac{1}{3}a$, $\frac{1}{9}a^{6}+\frac{1}{9}a^{4}-\frac{2}{9}a^{2}$, $\frac{1}{9}a^{7}+\frac{1}{9}a^{5}+\frac{1}{9}a^{3}-\frac{1}{3}a$, $\frac{1}{81}a^{8}-\frac{1}{27}a^{7}-\frac{1}{27}a^{6}-\frac{1}{9}a^{5}-\frac{2}{27}a^{4}-\frac{19}{81}a^{2}+\frac{4}{27}a+\frac{1}{3}$, $\frac{1}{243}a^{9}+\frac{1}{243}a^{8}-\frac{2}{81}a^{7}-\frac{4}{81}a^{6}+\frac{7}{81}a^{5}+\frac{13}{81}a^{4}-\frac{10}{243}a^{3}+\frac{26}{243}a^{2}-\frac{29}{81}a+\frac{4}{9}$, $\frac{1}{333153}a^{10}+\frac{40}{37017}a^{9}+\frac{2042}{333153}a^{8}+\frac{991}{111051}a^{7}-\frac{4909}{111051}a^{6}+\frac{4607}{37017}a^{5}+\frac{28535}{333153}a^{4}+\frac{58}{457}a^{3}-\frac{148340}{333153}a^{2}-\frac{10396}{111051}a-\frac{2662}{12339}$
Monogenic: | No | |
Index: | Not computed | |
Inessential primes: | $3$ |
Class group and class number
Trivial group, which has order $1$
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: | $\frac{54967}{333153}a^{10}+\frac{33520}{111051}a^{9}-\frac{2845981}{333153}a^{8}-\frac{2339201}{111051}a^{7}+\frac{10951910}{111051}a^{6}+\frac{3035437}{12339}a^{5}-\frac{127705516}{333153}a^{4}-\frac{101592145}{111051}a^{3}+\frac{161326672}{333153}a^{2}+\frac{114888410}{111051}a+\frac{278876}{12339}$, $\frac{1726}{111051}a^{10}+\frac{2384}{37017}a^{9}-\frac{96319}{111051}a^{8}-\frac{133529}{37017}a^{7}+\frac{387839}{37017}a^{6}+\frac{490243}{12339}a^{5}-\frac{4624777}{111051}a^{4}-\frac{5321174}{37017}a^{3}+\frac{6013987}{111051}a^{2}+\frac{5894081}{37017}a+\frac{14690}{4113}$, $\frac{6256}{37017}a^{10}+\frac{34010}{111051}a^{9}-\frac{970432}{111051}a^{8}-\frac{792793}{37017}a^{7}+\frac{3721810}{37017}a^{6}+\frac{9227072}{37017}a^{5}-\frac{14352293}{37017}a^{4}-\frac{102335030}{111051}a^{3}+\frac{53648395}{111051}a^{2}+\frac{38360489}{37017}a+\frac{102563}{4113}$, $\frac{5650}{111051}a^{10}+\frac{8576}{111051}a^{9}-\frac{292145}{111051}a^{8}-\frac{70078}{12339}a^{7}+\frac{42194}{1371}a^{6}+\frac{2425445}{37017}a^{5}-\frac{13490635}{111051}a^{4}-\frac{26463293}{111051}a^{3}+\frac{17362892}{111051}a^{2}+\frac{9818998}{37017}a-\frac{13226}{4113}$, $\frac{275}{12339}a^{10}+\frac{4420}{111051}a^{9}-\frac{127973}{111051}a^{8}-\frac{103733}{37017}a^{7}+\frac{491759}{37017}a^{6}+\frac{1217059}{37017}a^{5}-\frac{1899815}{37017}a^{4}-\frac{13750696}{111051}a^{3}+\frac{7173947}{111051}a^{2}+\frac{5330362}{37017}a+\frac{2404}{4113}$, $\frac{6076}{111051}a^{10}+\frac{10669}{111051}a^{9}-\frac{104620}{37017}a^{8}-\frac{250552}{37017}a^{7}+\frac{1203673}{37017}a^{6}+\frac{2898655}{37017}a^{5}-\frac{13909537}{111051}a^{4}-\frac{31813915}{111051}a^{3}+\frac{213575}{1371}a^{2}+\frac{3954112}{12339}a+\frac{8170}{1371}$, $\frac{35558}{333153}a^{10}+\frac{21430}{111051}a^{9}-\frac{1838606}{333153}a^{8}-\frac{1497967}{111051}a^{7}+\frac{7053922}{111051}a^{6}+\frac{5791961}{37017}a^{5}-\frac{81878270}{333153}a^{4}-\frac{63789862}{111051}a^{3}+\frac{103533065}{333153}a^{2}+\frac{70867381}{111051}a+\frac{154738}{12339}$, $\frac{319529}{333153}a^{10}+\frac{190378}{111051}a^{9}-\frac{16536878}{333153}a^{8}-\frac{13368397}{111051}a^{7}+\frac{63699496}{111051}a^{6}+\frac{51964082}{37017}a^{5}-\frac{742799654}{333153}a^{4}-\frac{578613064}{111051}a^{3}+\frac{935356439}{333153}a^{2}+\frac{655012909}{111051}a+\frac{1591756}{12339}$, $\frac{34499}{111051}a^{10}+\frac{21853}{37017}a^{9}-\frac{1782806}{111051}a^{8}-\frac{1508218}{37017}a^{7}+\frac{6789118}{37017}a^{6}+\frac{5847377}{12339}a^{5}-\frac{77617556}{111051}a^{4}-\frac{64751773}{37017}a^{3}+\frac{94804046}{111051}a^{2}+\frac{72673222}{37017}a+\frac{309943}{4113}$, $\frac{17357}{111051}a^{10}+\frac{10345}{37017}a^{9}-\frac{898103}{111051}a^{8}-\frac{726409}{37017}a^{7}+\frac{3457057}{37017}a^{6}+\frac{2823470}{12339}a^{5}-\frac{40251170}{111051}a^{4}-\frac{31447798}{37017}a^{3}+\frac{50523341}{111051}a^{2}+\frac{35704375}{37017}a+\frac{113530}{4113}$ | 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: | \( 278432564.723 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{0}\cdot 278432564.723 \cdot 1}{2\cdot\sqrt{672749994932560009201}}\cr\approx \mathstrut & 10.9924154253 \end{aligned}\]
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 | ${\href{/padicField/2.11.0.1}{11} }$ | ${\href{/padicField/3.1.0.1}{1} }^{11}$ | ${\href{/padicField/5.11.0.1}{11} }$ | ${\href{/padicField/7.11.0.1}{11} }$ | R | ${\href{/padicField/13.11.0.1}{11} }$ | ${\href{/padicField/17.11.0.1}{11} }$ | ${\href{/padicField/19.11.0.1}{11} }$ | ${\href{/padicField/23.11.0.1}{11} }$ | ${\href{/padicField/29.11.0.1}{11} }$ | ${\href{/padicField/31.11.0.1}{11} }$ | ${\href{/padicField/37.11.0.1}{11} }$ | ${\href{/padicField/41.11.0.1}{11} }$ | ${\href{/padicField/43.11.0.1}{11} }$ | ${\href{/padicField/47.11.0.1}{11} }$ | ${\href{/padicField/53.11.0.1}{11} }$ | ${\href{/padicField/59.11.0.1}{11} }$ |
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 |
---|---|---|---|---|---|---|---|
\(11\) | 11.11.20.9 | $x^{11} + 110 x^{10} + 11$ | $11$ | $1$ | $20$ | $C_{11}$ | $[2]$ |