Normalized defining polynomial
\( x^{11} - 2x^{10} + x^{9} + 4x^{8} - 6x^{7} + 2x^{6} + 5x^{5} - 5x^{4} + 3x^{2} - 3x + 1 \)
Invariants
Degree: | $11$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[3, 4]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(118125032125\) \(\medspace = 5^{3}\cdot 73\cdot 139\cdot 93131\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(10.15\) | 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: | $5^{3/4}73^{1/2}139^{1/2}93131^{1/2}\approx 102788.2486816166$ | ||
Ramified primes: | \(5\), \(73\), \(139\), \(93131\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | $\Q(\sqrt{4725001285}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $6$ | 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)
| |
Fundamental units: | $a$, $4a^{10}-5a^{9}+16a^{7}-11a^{6}-2a^{5}+18a^{4}-4a^{3}-5a^{2}+7a-5$, $a^{10}-a^{9}+4a^{7}-2a^{6}+5a^{4}-a^{3}+3a-2$, $2a^{10}-3a^{9}+9a^{7}-8a^{6}-2a^{5}+12a^{4}-5a^{3}-5a^{2}+6a-3$, $4a^{10}-6a^{9}+a^{8}+17a^{7}-16a^{6}+22a^{4}-10a^{3}-6a^{2}+11a-6$, $5a^{10}-7a^{9}+a^{8}+21a^{7}-18a^{6}+26a^{4}-10a^{3}-5a^{2}+13a-8$ | 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: | \( 12.5932309979 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{4}\cdot 12.5932309979 \cdot 1}{2\cdot\sqrt{118125032125}}\cr\approx \mathstrut & 0.228426068280 \end{aligned}\]
Galois group
A non-solvable group of order 39916800 |
The 56 conjugacy class representatives for $S_{11}$ |
Character table for $S_{11}$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 22 sibling: | data not computed |
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 | ${\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.3.0.1}{3} }$ | ${\href{/padicField/3.5.0.1}{5} }{,}\,{\href{/padicField/3.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.11.0.1}{11} }$ | ${\href{/padicField/13.11.0.1}{11} }$ | ${\href{/padicField/17.11.0.1}{11} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ | ${\href{/padicField/29.11.0.1}{11} }$ | ${\href{/padicField/31.11.0.1}{11} }$ | ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.11.0.1}{11} }$ | ${\href{/padicField/43.7.0.1}{7} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(5\) | 5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
5.4.0.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | |
5.4.3.1 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $[\ ]_{4}$ | |
\(73\) | $\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{73}$ | $x + 68$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
73.2.1.1 | $x^{2} + 73$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
73.2.0.1 | $x^{2} + 70 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
73.3.0.1 | $x^{3} + 2 x + 68$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
\(139\) | $\Q_{139}$ | $x + 137$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{139}$ | $x + 137$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
139.2.0.1 | $x^{2} + 138 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
139.2.1.1 | $x^{2} + 278$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
139.5.0.1 | $x^{5} + 10 x + 137$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | |
\(93131\) | $\Q_{93131}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{93131}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{93131}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |