Normalized defining polynomial
\( x^{11} - x^{10} - 30 x^{9} + 63 x^{8} + 220 x^{7} - 698 x^{6} - 101 x^{5} + 1960 x^{4} - 1758 x^{3} + 35 x^{2} + 243 x + 29 \)
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: | \(1822837804551761449\) \(\medspace = 67^{10}\) | 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: | \(45.72\) | 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: | $67^{10/11}\approx 45.71597667454237$ | ||
Ramified primes: | \(67\) | 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: | \(67\) | ||
Dirichlet character group: | $\lbrace$$\chi_{67}(64,·)$, $\chi_{67}(1,·)$, $\chi_{67}(40,·)$, $\chi_{67}(9,·)$, $\chi_{67}(14,·)$, $\chi_{67}(15,·)$, $\chi_{67}(22,·)$, $\chi_{67}(24,·)$, $\chi_{67}(25,·)$, $\chi_{67}(59,·)$, $\chi_{67}(62,·)$$\rbrace$ | ||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{256447}a^{10}-\frac{92780}{256447}a^{9}-\frac{120859}{256447}a^{8}+\frac{32149}{256447}a^{7}-\frac{16794}{256447}a^{6}-\frac{42144}{256447}a^{5}+\frac{30666}{256447}a^{4}+\frac{4159}{8843}a^{3}-\frac{104882}{256447}a^{2}-\frac{34302}{256447}a-\frac{61}{8843}$
Monogenic: | No | |
Index: | $1$ | |
Inessential primes: | None |
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)
| |
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)
| |
Fundamental units: | $\frac{347046}{256447}a^{10}+\frac{300993}{256447}a^{9}-\frac{9931968}{256447}a^{8}+\frac{3276036}{256447}a^{7}+\frac{84864402}{256447}a^{6}-\frac{85361724}{256447}a^{5}-\frac{214427556}{256447}a^{4}+\frac{10577239}{8843}a^{3}+\frac{6337548}{256447}a^{2}-\frac{63444561}{256447}a-\frac{300326}{8843}$, $\frac{202849}{256447}a^{10}+\frac{58663}{256447}a^{9}-\frac{5948819}{256447}a^{8}+\frac{5074231}{256447}a^{7}+\frac{49233666}{256447}a^{6}-\frac{75346482}{256447}a^{5}-\frac{101107563}{256447}a^{4}+\frac{8109293}{8843}a^{3}-\frac{87044784}{256447}a^{2}-\frac{11490062}{256447}a+\frac{41783}{8843}$, $\frac{396984}{256447}a^{10}+\frac{281302}{256447}a^{9}-\frac{11447247}{256447}a^{8}+\frac{5426154}{256447}a^{7}+\frac{97096776}{256447}a^{6}-\frac{111189414}{256447}a^{5}-\frac{234020304}{256447}a^{4}+\frac{13155996}{8843}a^{3}-\frac{37182230}{256447}a^{2}-\frac{62582536}{256447}a-\frac{251494}{8843}$, $\frac{2881780}{256447}a^{10}+\frac{2145376}{256447}a^{9}-\frac{82747950}{256447}a^{8}+\frac{37178792}{256447}a^{7}+\frac{699908383}{256447}a^{6}-\frac{791166373}{256447}a^{5}-\frac{1679980805}{256447}a^{4}+\frac{94114391}{8843}a^{3}-\frac{286149294}{256447}a^{2}-\frac{436716840}{256447}a-\frac{1705282}{8843}$, $\frac{648138}{256447}a^{10}+\frac{526284}{256447}a^{9}-\frac{18499894}{256447}a^{8}+\frac{7337434}{256447}a^{7}+\frac{156239566}{256447}a^{6}-\frac{170212922}{256447}a^{5}-\frac{377101917}{256447}a^{4}+\frac{20545384}{8843}a^{3}-\frac{53405720}{256447}a^{2}-\frac{98489106}{256447}a-\frac{397300}{8843}$, $\frac{793968}{256447}a^{10}+\frac{562604}{256447}a^{9}-\frac{22894494}{256447}a^{8}+\frac{10852308}{256447}a^{7}+\frac{194193552}{256447}a^{6}-\frac{222378828}{256447}a^{5}-\frac{468040608}{256447}a^{4}+\frac{26311992}{8843}a^{3}-\frac{74108013}{256447}a^{2}-\frac{124652178}{256447}a-\frac{502988}{8843}$, $a^{10}+a^{9}-28a^{8}+7a^{7}+234a^{6}-230a^{5}-561a^{4}+838a^{3}-82a^{2}-129a-15$, $\frac{1038950}{256447}a^{10}+\frac{799595}{256447}a^{9}-\frac{29753269}{256447}a^{8}+\frac{12829938}{256447}a^{7}+\frac{251332746}{256447}a^{6}-\frac{281070379}{256447}a^{5}-\frac{602928583}{256447}a^{4}+\frac{33588302}{8843}a^{3}-\frac{102581483}{256447}a^{2}-\frac{155286639}{256447}a-\frac{617179}{8843}$, $\frac{659504}{256447}a^{10}+\frac{498868}{256447}a^{9}-\frac{18909603}{256447}a^{8}+\frac{8331781}{256447}a^{7}+\frac{159745788}{256447}a^{6}-\frac{179667169}{256447}a^{5}-\frac{382449021}{256447}a^{4}+\frac{21390828}{8843}a^{3}-\frac{68402802}{256447}a^{2}-\frac{96258175}{256447}a-\frac{365500}{8843}$, $\frac{239689}{256447}a^{10}+\frac{225526}{256447}a^{9}-\frac{6730906}{256447}a^{8}+\frac{2093781}{256447}a^{7}+\frac{56273386}{256447}a^{6}-\frac{58732249}{256447}a^{5}-\frac{134102821}{256447}a^{4}+\frac{7255264}{8843}a^{3}-\frac{23668306}{256447}a^{2}-\frac{33459368}{256447}a-\frac{127352}{8843}$ | 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: | \( 330512.248088 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{11}\cdot(2\pi)^{0}\cdot 330512.248088 \cdot 1}{2\cdot\sqrt{1822837804551761449}}\cr\approx \mathstrut & 0.250676430123 \end{aligned}\]
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 | ${\href{/padicField/2.11.0.1}{11} }$ | ${\href{/padicField/3.11.0.1}{11} }$ | ${\href{/padicField/5.11.0.1}{11} }$ | ${\href{/padicField/7.11.0.1}{11} }$ | ${\href{/padicField/11.11.0.1}{11} }$ | ${\href{/padicField/13.11.0.1}{11} }$ | ${\href{/padicField/17.11.0.1}{11} }$ | ${\href{/padicField/19.11.0.1}{11} }$ | ${\href{/padicField/23.11.0.1}{11} }$ | ${\href{/padicField/29.1.0.1}{1} }^{11}$ | ${\href{/padicField/31.11.0.1}{11} }$ | ${\href{/padicField/37.1.0.1}{1} }^{11}$ | ${\href{/padicField/41.11.0.1}{11} }$ | ${\href{/padicField/43.11.0.1}{11} }$ | ${\href{/padicField/47.11.0.1}{11} }$ | ${\href{/padicField/53.11.0.1}{11} }$ | ${\href{/padicField/59.11.0.1}{11} }$ |
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 |
---|---|---|---|---|---|---|---|
\(67\) | 67.11.10.1 | $x^{11} + 67$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ |