Normalized defining polynomial
\( x^{11} - 5x^{10} + 13x^{9} - 22x^{8} + 32x^{7} - 44x^{6} + 47x^{5} - 30x^{4} + 11x^{3} + 6x^{2} - 21x + 17 \)
Invariants
Degree: | $11$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[1, 5]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-93851287159343\) \(\medspace = -\,7^{5}\cdot 89^{5}\) | 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: | \(18.63\) | 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: | $7^{1/2}89^{1/2}\approx 24.95996794869737$ | ||
Ramified primes: | \(7\), \(89\) | 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: | \(\Q(\sqrt{-623}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{7}a^{8}+\frac{1}{7}a^{7}+\frac{2}{7}a^{5}+\frac{3}{7}a^{4}-\frac{1}{7}a^{3}-\frac{1}{7}a-\frac{3}{7}$, $\frac{1}{7}a^{9}-\frac{1}{7}a^{7}+\frac{2}{7}a^{6}+\frac{1}{7}a^{5}+\frac{3}{7}a^{4}+\frac{1}{7}a^{3}-\frac{1}{7}a^{2}-\frac{2}{7}a+\frac{3}{7}$, $\frac{1}{49}a^{10}-\frac{3}{49}a^{9}-\frac{15}{49}a^{7}+\frac{16}{49}a^{6}+\frac{23}{49}a^{5}+\frac{2}{49}a^{4}+\frac{23}{49}a^{3}-\frac{6}{49}a^{2}+\frac{1}{49}a-\frac{12}{49}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $5$ | 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: | $\frac{1}{49}a^{10}+\frac{4}{49}a^{9}-\frac{1}{7}a^{8}+\frac{20}{49}a^{7}-\frac{19}{49}a^{6}+\frac{65}{49}a^{5}-\frac{47}{49}a^{4}+\frac{37}{49}a^{3}-\frac{62}{49}a^{2}+\frac{43}{49}a+\frac{128}{49}$, $\frac{18}{49}a^{10}-\frac{89}{49}a^{9}+4a^{8}-\frac{284}{49}a^{7}+\frac{365}{49}a^{6}-\frac{552}{49}a^{5}+\frac{470}{49}a^{4}-\frac{111}{49}a^{3}-\frac{24}{49}a^{2}+\frac{39}{49}a-\frac{321}{49}$, $\frac{15}{49}a^{10}-\frac{59}{49}a^{9}+\frac{18}{7}a^{8}-\frac{183}{49}a^{7}+\frac{261}{49}a^{6}-\frac{348}{49}a^{5}+\frac{268}{49}a^{4}-\frac{89}{49}a^{3}+\frac{22}{49}a^{2}+\frac{162}{49}a-\frac{208}{49}$, $\frac{40}{49}a^{10}-\frac{169}{49}a^{9}+\frac{52}{7}a^{8}-\frac{530}{49}a^{7}+\frac{738}{49}a^{6}-\frac{1047}{49}a^{5}+\frac{829}{49}a^{4}-\frac{326}{49}a^{3}+\frac{103}{49}a^{2}+\frac{264}{49}a-\frac{543}{49}$, $\frac{1}{49}a^{10}-\frac{3}{49}a^{9}+\frac{1}{7}a^{8}-\frac{8}{49}a^{7}+\frac{16}{49}a^{6}-\frac{12}{49}a^{5}+\frac{23}{49}a^{4}-\frac{33}{49}a^{3}+\frac{43}{49}a^{2}-\frac{55}{49}a+\frac{65}{49}$ | 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: | \( 582.446099571 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{5}\cdot 582.446099571 \cdot 1}{2\cdot\sqrt{93851287159343}}\cr\approx \mathstrut & 0.588755485387 \end{aligned}\]
Galois group
A solvable group of order 22 |
The 7 conjugacy class representatives for $D_{11}$ |
Character table for $D_{11}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Galois closure: | deg 22 |
Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | ${\href{/padicField/2.11.0.1}{11} }$ | ${\href{/padicField/3.11.0.1}{11} }$ | ${\href{/padicField/5.2.0.1}{2} }^{5}{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | ${\href{/padicField/11.11.0.1}{11} }$ | ${\href{/padicField/13.11.0.1}{11} }$ | ${\href{/padicField/17.2.0.1}{2} }^{5}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.11.0.1}{11} }$ | ${\href{/padicField/23.2.0.1}{2} }^{5}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.2.0.1}{2} }^{5}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.11.0.1}{11} }$ | ${\href{/padicField/37.2.0.1}{2} }^{5}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.11.0.1}{11} }$ | ${\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.2.0.1}{2} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.11.0.1}{11} }$ | ${\href{/padicField/59.11.0.1}{11} }$ |
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 |
---|---|---|---|---|---|---|---|
\(7\) | $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
7.2.1.1 | $x^{2} + 21$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
\(89\) | $\Q_{89}$ | $x + 86$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
89.2.1.2 | $x^{2} + 267$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
89.2.1.2 | $x^{2} + 267$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
89.2.1.2 | $x^{2} + 267$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
89.2.1.2 | $x^{2} + 267$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
89.2.1.2 | $x^{2} + 267$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |