Normalized defining polynomial
\( x^{11} - 2x^{10} - x^{9} + 5x^{8} - 3x^{7} - 2x^{6} - 3x^{5} + 3x^{4} + 11x^{3} - 5x^{2} - 4x + 1 \)
Invariants
Degree: | $11$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[7, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(1135258109569\) \(\medspace = 71\cdot 73\cdot 219034943\) | 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: | \(12.47\) | 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: | $71^{1/2}73^{1/2}219034943^{1/2}\approx 1065484.9175699297$ | ||
Ramified primes: | \(71\), \(73\), \(219034943\) | 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: | $\Q(\sqrt{1135258109569}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{53}a^{10}-\frac{14}{53}a^{9}+\frac{8}{53}a^{8}+\frac{15}{53}a^{7}-\frac{24}{53}a^{6}+\frac{21}{53}a^{5}+\frac{10}{53}a^{4}-\frac{11}{53}a^{3}-\frac{16}{53}a^{2}-\frac{25}{53}a-\frac{22}{53}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $8$ | 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: | $\frac{44}{53}a^{10}-\frac{33}{53}a^{9}-\frac{72}{53}a^{8}+\frac{130}{53}a^{7}+\frac{4}{53}a^{6}-\frac{83}{53}a^{5}-\frac{196}{53}a^{4}-\frac{113}{53}a^{3}+\frac{250}{53}a^{2}+\frac{13}{53}a-\frac{120}{53}$, $a$, $\frac{105}{53}a^{10}-\frac{92}{53}a^{9}-\frac{167}{53}a^{8}+\frac{303}{53}a^{7}-\frac{29}{53}a^{6}-\frac{127}{53}a^{5}-\frac{487}{53}a^{4}-\frac{254}{53}a^{3}+\frac{652}{53}a^{2}+\frac{78}{53}a-\frac{84}{53}$, $\frac{24}{53}a^{10}+\frac{35}{53}a^{9}-\frac{73}{53}a^{8}-\frac{11}{53}a^{7}+\frac{113}{53}a^{6}-\frac{26}{53}a^{5}-\frac{131}{53}a^{4}-\frac{370}{53}a^{3}-\frac{66}{53}a^{2}+\frac{248}{53}a+\frac{55}{53}$, $\frac{34}{53}a^{10}+\frac{1}{53}a^{9}-\frac{46}{53}a^{8}+\frac{33}{53}a^{7}+\frac{32}{53}a^{6}+\frac{25}{53}a^{5}-\frac{190}{53}a^{4}-\frac{268}{53}a^{3}-\frac{14}{53}a^{2}+\frac{104}{53}a+\frac{47}{53}$, $\frac{49}{53}a^{10}-\frac{50}{53}a^{9}-\frac{85}{53}a^{8}+\frac{152}{53}a^{7}-\frac{10}{53}a^{6}-\frac{84}{53}a^{5}-\frac{252}{53}a^{4}-\frac{62}{53}a^{3}+\frac{382}{53}a^{2}+\frac{47}{53}a-\frac{71}{53}$, $\frac{11}{53}a^{10}+\frac{5}{53}a^{9}-\frac{18}{53}a^{8}+\frac{6}{53}a^{7}+\frac{1}{53}a^{6}+\frac{19}{53}a^{5}-\frac{49}{53}a^{4}-\frac{121}{53}a^{3}-\frac{17}{53}a^{2}-\frac{10}{53}a+\frac{76}{53}$, $\frac{10}{53}a^{10}+\frac{72}{53}a^{9}-\frac{79}{53}a^{8}-\frac{62}{53}a^{7}+\frac{184}{53}a^{6}-\frac{55}{53}a^{5}-\frac{59}{53}a^{4}-\frac{428}{53}a^{3}-\frac{160}{53}a^{2}+\frac{280}{53}a-\frac{8}{53}$ | 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: | \( 72.38649544528622 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{2}\cdot 72.38649544528622 \cdot 1}{2\cdot\sqrt{1135258109569}}\cr\approx \mathstrut & 0.171652429738605 \end{aligned}\]
Galois group
A non-solvable group of order 39916800 |
The 56 conjugacy class representatives for $S_{11}$ are not computed |
Character table for $S_{11}$ is not computed |
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 | ${\href{/padicField/2.11.0.1}{11} }$ | ${\href{/padicField/3.11.0.1}{11} }$ | ${\href{/padicField/5.11.0.1}{11} }$ | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.11.0.1}{11} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.9.0.1}{9} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.3.0.1}{3} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.11.0.1}{11} }$ | ${\href{/padicField/41.7.0.1}{7} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.11.0.1}{11} }$ | ${\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
---|---|---|---|---|---|---|---|
\(71\) | 71.2.1.1 | $x^{2} + 497$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
71.3.0.1 | $x^{3} + 4 x + 64$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
71.6.0.1 | $x^{6} + x^{4} + 10 x^{3} + 13 x^{2} + 29 x + 7$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | |
\(73\) | 73.2.0.1 | $x^{2} + 70 x + 5$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
73.2.1.2 | $x^{2} + 365$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
73.7.0.1 | $x^{7} + 10 x + 68$ | $1$ | $7$ | $0$ | $C_7$ | $[\ ]^{7}$ | |
\(219034943\) | $\Q_{219034943}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{219034943}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ |