Normalized defining polynomial
\( x^{22} - 2 x^{20} - 2 x^{19} + 5 x^{18} + 6 x^{17} - 5 x^{16} - 12 x^{15} + 2 x^{14} + 18 x^{13} + \cdots + 1 \)
Invariants
Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[0, 11]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(-1441291887151686135870128128\) \(\medspace = -\,2^{22}\cdot 10177\cdot 33765428299628041\) | 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: | \(17.16\) | 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: | not computed | ||
Ramified primes: | \(2\), \(10177\), \(33765428299628041\) | 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: | $\Q(\sqrt{-34363\!\cdots\!73257}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$
Monogenic: | Yes | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $10$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( 2 a^{20} - 3 a^{18} - 4 a^{17} + 8 a^{16} + 10 a^{15} - 5 a^{14} - 18 a^{13} - a^{12} + 24 a^{11} + 10 a^{10} - 18 a^{9} - 18 a^{8} + 11 a^{7} + 19 a^{6} - 3 a^{5} - 10 a^{4} - 2 a^{3} + 6 a^{2} + a - 2 \) (order $4$) | 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: | $a^{21}-2a^{19}-2a^{18}+5a^{17}+6a^{16}-5a^{15}-12a^{14}+2a^{13}+18a^{12}+4a^{11}-18a^{10}-11a^{9}+14a^{8}+15a^{7}-8a^{6}-12a^{5}+2a^{4}+8a^{3}-4a$, $a^{21}-a^{20}-a^{19}-a^{18}+6a^{17}+a^{16}-6a^{15}-7a^{14}+8a^{13}+13a^{12}-6a^{11}-14a^{10}-a^{9}+16a^{8}+5a^{7}-13a^{6}-4a^{5}+4a^{4}+6a^{3}-4a^{2}-3a+2$, $a^{21}+a^{20}-2a^{19}-4a^{18}+2a^{17}+11a^{16}+2a^{15}-16a^{14}-13a^{13}+17a^{12}+23a^{11}-10a^{10}-29a^{9}-3a^{8}+26a^{7}+10a^{6}-17a^{5}-11a^{4}+6a^{3}+7a^{2}-3a-3$, $a^{21}-2a^{19}-2a^{18}+5a^{17}+6a^{16}-5a^{15}-12a^{14}+2a^{13}+18a^{12}+4a^{11}-18a^{10}-11a^{9}+14a^{8}+15a^{7}-8a^{6}-11a^{5}+2a^{4}+7a^{3}-3a+1$, $a^{19}-a^{17}-2a^{16}+3a^{15}+4a^{14}-6a^{12}-3a^{11}+6a^{10}+6a^{9}-a^{8}-7a^{7}-2a^{6}+5a^{5}+3a^{4}-3a^{2}$, $a^{20}-2a^{18}-2a^{17}+5a^{16}+5a^{15}-5a^{14}-11a^{13}+3a^{12}+15a^{11}+a^{10}-16a^{9}-8a^{8}+13a^{7}+9a^{6}-7a^{5}-9a^{4}+3a^{3}+5a^{2}-a-2$, $a^{20}-a^{18}-3a^{17}+3a^{16}+6a^{15}+a^{14}-11a^{13}-7a^{12}+12a^{11}+13a^{10}-6a^{9}-19a^{8}-a^{7}+16a^{6}+5a^{5}-8a^{4}-8a^{3}+4a^{2}+3a-1$, $2a^{21}+a^{20}-3a^{19}-6a^{18}+7a^{17}+15a^{16}-a^{15}-24a^{14}-8a^{13}+30a^{12}+22a^{11}-21a^{10}-30a^{9}+11a^{8}+30a^{7}+a^{6}-16a^{5}-5a^{4}+9a^{3}+3a^{2}-2a$, $a^{21}+2a^{20}-2a^{19}-5a^{18}+a^{17}+13a^{16}+5a^{15}-15a^{14}-15a^{13}+13a^{12}+24a^{11}-4a^{10}-23a^{9}-6a^{8}+17a^{7}+10a^{6}-8a^{5}-6a^{4}+a^{3}+2a^{2}-a-1$, $a^{21}-a^{20}-2a^{19}-a^{18}+6a^{17}+3a^{16}-8a^{15}-10a^{14}+5a^{13}+17a^{12}-2a^{11}-17a^{10}-8a^{9}+14a^{8}+12a^{7}-9a^{6}-8a^{5}+a^{4}+6a^{3}-a^{2}-2a+1$ (assuming GRH) | 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: | \( 148004.116661 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 148004.116661 \cdot 1}{4\cdot\sqrt{1441291887151686135870128128}}\cr\approx \mathstrut & 0.587241020674 \end{aligned}\] (assuming GRH)
Galois group
$S_{11}\wr C_2$ (as 22T57):
A non-solvable group of order 3186701844480000 |
The 1652 conjugacy class representatives for $S_{11}\wr C_2$ are not computed |
Character table for $S_{11}\wr C_2$ is not computed |
Intermediate fields
\(\Q(\sqrt{-1}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 44 siblings: | 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 | R | $22$ | ${\href{/padicField/5.9.0.1}{9} }{,}\,{\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | $16{,}\,{\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.12.0.1}{12} }{,}\,{\href{/padicField/11.10.0.1}{10} }$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.7.0.1}{7} }{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $22$ | ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.4.0.1}{4} }$ | ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.14.0.1}{14} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.7.0.1}{7} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }^{4}$ | ${\href{/padicField/53.11.0.1}{11} }{,}\,{\href{/padicField/53.3.0.1}{3} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
2.6.6.3 | $x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ | |
2.14.14.38 | $x^{14} + 14 x^{13} + 98 x^{12} + 448 x^{11} + 1948 x^{10} + 8392 x^{9} + 30520 x^{8} + 84992 x^{7} + 178608 x^{6} + 284064 x^{5} + 325984 x^{4} + 242688 x^{3} + 97600 x^{2} + 11648 x - 5504$ | $2$ | $7$ | $14$ | $C_{14}$ | $[2]^{7}$ | |
\(10177\) | $\Q_{10177}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{10177}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{10177}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{10177}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
\(33765428299628041\) | $\Q_{33765428299628041}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | ||
Deg $6$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ | ||
Deg $8$ | $1$ | $8$ | $0$ | $C_8$ | $[\ ]^{8}$ |