Normalized defining polynomial
\( x^{22} - x^{21} + 3 x^{20} - 7 x^{19} + 11 x^{18} - 18 x^{17} + 29 x^{16} - 40 x^{15} + 53 x^{14} - 67 x^{13} + 79 x^{12} - 87 x^{11} + 92 x^{10} - 90 x^{9} + 83 x^{8} - 73 x^{7} + 59 x^{6} - 43 x^{5} + \cdots + 1 \)
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: | \(-64196743029301100734358828\) \(\medspace = -\,2^{2}\cdot 2029\cdot 7909899338257898069783\) | 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: | \(14.90\) | 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: | $2^{2/3}2029^{1/2}7909899338257898069783^{1/2}\approx 6359356409154.484$ | ||
Ramified primes: | \(2\), \(2029\), \(7909899338257898069783\) | 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: | $\Q(\sqrt{-16049\!\cdots\!89707}$) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | 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: | \( -1 \) (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: | $a^{21}-a^{20}+3a^{19}-7a^{18}+11a^{17}-18a^{16}+29a^{15}-40a^{14}+53a^{13}-67a^{12}+79a^{11}-87a^{10}+92a^{9}-90a^{8}+83a^{7}-73a^{6}+59a^{5}-43a^{4}+30a^{3}-18a^{2}+9a-4$, $3a^{21}-a^{20}+9a^{19}-16a^{18}+24a^{17}-43a^{16}+66a^{15}-87a^{14}+119a^{13}-146a^{12}+169a^{11}-184a^{10}+194a^{9}-182a^{8}+168a^{7}-145a^{6}+112a^{5}-80a^{4}+57a^{3}-29a^{2}+15a-6$, $a^{21}-a^{20}+4a^{19}-7a^{18}+13a^{17}-22a^{16}+33a^{15}-47a^{14}+64a^{13}-78a^{12}+93a^{11}-102a^{10}+106a^{9}-102a^{8}+94a^{7}-79a^{6}+62a^{5}-45a^{4}+29a^{3}-15a^{2}+8a-2$, $a^{4}-a+1$, $a^{21}-a^{20}+3a^{19}-7a^{18}+11a^{17}-18a^{16}+29a^{15}-40a^{14}+53a^{13}-67a^{12}+79a^{11}-87a^{10}+92a^{9}-90a^{8}+83a^{7}-73a^{6}+59a^{5}-42a^{4}+30a^{3}-17a^{2}+8a-3$, $2a^{21}+5a^{19}-9a^{18}+11a^{17}-21a^{16}+33a^{15}-40a^{14}+55a^{13}-68a^{12}+77a^{11}-83a^{10}+89a^{9}-83a^{8}+78a^{7}-69a^{6}+54a^{5}-39a^{4}+30a^{3}-15a^{2}+8a-4$, $a^{21}-a^{20}+3a^{19}-7a^{18}+11a^{17}-18a^{16}+29a^{15}-40a^{14}+53a^{13}-67a^{12}+79a^{11}-87a^{10}+92a^{9}-90a^{8}+82a^{7}-73a^{6}+58a^{5}-41a^{4}+29a^{3}-16a^{2}+7a-3$, $3a^{21}+9a^{19}-13a^{18}+20a^{17}-36a^{16}+55a^{15}-70a^{14}+97a^{13}-117a^{12}+136a^{11}-145a^{10}+154a^{9}-142a^{8}+132a^{7}-111a^{6}+86a^{5}-59a^{4}+43a^{3}-19a^{2}+11a-3$, $4a^{21}-a^{20}+11a^{19}-20a^{18}+28a^{17}-50a^{16}+78a^{15}-99a^{14}+134a^{13}-165a^{12}+188a^{11}-201a^{10}+212a^{9}-196a^{8}+178a^{7}-153a^{6}+116a^{5}-80a^{4}+57a^{3}-27a^{2}+13a-6$, $2a^{21}+5a^{19}-9a^{18}+11a^{17}-21a^{16}+33a^{15}-40a^{14}+54a^{13}-68a^{12}+75a^{11}-80a^{10}+85a^{9}-78a^{8}+70a^{7}-61a^{6}+46a^{5}-31a^{4}+23a^{3}-11a^{2}+5a-3$ (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)]
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Regulator: | \( 9202.86473748 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{11}\cdot 9202.86473748 \cdot 1}{2\cdot\sqrt{64196743029301100734358828}}\cr\approx \mathstrut & 0.346031071546 \end{aligned}\] (assuming GRH)
Galois group
A non-solvable group of order 1124000727777607680000 |
The 1002 conjugacy class representatives for $S_{22}$ are not computed |
Character table for $S_{22}$ is not computed |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 44 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 | R | $19{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.11.0.1}{11} }{,}\,{\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ | $19{,}\,{\href{/padicField/7.3.0.1}{3} }$ | ${\href{/padicField/11.11.0.1}{11} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.3.0.1}{3} }$ | ${\href{/padicField/13.13.0.1}{13} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | $20{,}\,{\href{/padicField/17.2.0.1}{2} }$ | $17{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ | $20{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.11.0.1}{11} }{,}\,{\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | $16{,}\,{\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.10.0.1}{10} }{,}\,{\href{/padicField/37.5.0.1}{5} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | $18{,}\,{\href{/padicField/43.4.0.1}{4} }$ | ${\href{/padicField/47.14.0.1}{14} }{,}\,{\href{/padicField/47.7.0.1}{7} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ | ${\href{/padicField/53.9.0.1}{9} }{,}\,{\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.5.0.1}{5} }$ | ${\href{/padicField/59.14.0.1}{14} }{,}\,{\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
---|---|---|---|---|---|---|---|
\(2\) | 2.3.2.1 | $x^{3} + 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ |
2.19.0.1 | $x^{19} + x^{5} + x^{2} + x + 1$ | $1$ | $19$ | $0$ | $C_{19}$ | $[\ ]^{19}$ | |
\(2029\) | $\Q_{2029}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{2029}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $[\ ]^{5}$ | ||
Deg $13$ | $1$ | $13$ | $0$ | $C_{13}$ | $[\ ]^{13}$ | ||
\(790\!\cdots\!783\) | Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
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 $4$ | $1$ | $4$ | $0$ | $C_4$ | $[\ ]^{4}$ | ||
Deg $9$ | $1$ | $9$ | $0$ | $C_9$ | $[\ ]^{9}$ |